From f084f03f0865ae19f7f223ba826da27d45744604 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Mon, 23 Nov 2020 13:24:23 -0500 Subject: [PATCH 01/20] Migrate Inverse draft from Notion into Markdown --- content/matrices/content.md | 288 +++++++++++++++++- .../Screenshot_2020-11-23_at_12.16.18.png | Bin 0 -> 119232 bytes .../matrices/images/proto-4/Untitled 1.png | Bin 0 -> 2906 bytes .../matrices/images/proto-4/Untitled 10.png | Bin 0 -> 2822 bytes .../matrices/images/proto-4/Untitled 11.png | Bin 0 -> 2883 bytes .../matrices/images/proto-4/Untitled 2.png | Bin 0 -> 3185 bytes .../matrices/images/proto-4/Untitled 3.png | Bin 0 -> 13914 bytes .../matrices/images/proto-4/Untitled 4.png | Bin 0 -> 36929 bytes .../matrices/images/proto-4/Untitled 5.png | Bin 0 -> 53730 bytes .../matrices/images/proto-4/Untitled 6.png | Bin 0 -> 24248 bytes .../matrices/images/proto-4/Untitled 7.png | Bin 0 -> 5681 bytes .../matrices/images/proto-4/Untitled 8.png | Bin 0 -> 11272 bytes .../matrices/images/proto-4/Untitled 9.png | Bin 0 -> 14996 bytes content/matrices/images/proto-4/Untitled.png | Bin 0 -> 2168 bytes 14 files changed, 287 insertions(+), 1 deletion(-) create mode 100755 content/matrices/images/proto-4/Screenshot_2020-11-23_at_12.16.18.png create mode 100755 content/matrices/images/proto-4/Untitled 1.png create mode 100755 content/matrices/images/proto-4/Untitled 10.png create mode 100755 content/matrices/images/proto-4/Untitled 11.png create mode 100755 content/matrices/images/proto-4/Untitled 2.png create mode 100755 content/matrices/images/proto-4/Untitled 3.png create mode 100755 content/matrices/images/proto-4/Untitled 4.png create mode 100755 content/matrices/images/proto-4/Untitled 5.png create mode 100755 content/matrices/images/proto-4/Untitled 6.png create mode 100755 content/matrices/images/proto-4/Untitled 7.png create mode 100755 content/matrices/images/proto-4/Untitled 8.png create mode 100755 content/matrices/images/proto-4/Untitled 9.png create mode 100755 content/matrices/images/proto-4/Untitled.png diff --git a/content/matrices/content.md b/content/matrices/content.md index f11a033aa..23e75b148 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1183,8 +1183,294 @@ Let's see why this is true geometrically. > section: inverses > sectionStatus: dev +> id: inverses -{.todo} COMING SOON +### Intro: Inverse as the reverse of a transformation + +Every 2x2 matrix can be thought of as a linear transformation. What does it mean to "undo" a linear transformation? + +{.fixme} Have a more interesting example here, why undoing a transformation is useful? + +**Examples/Interactive:** Each of these transformations has a respective transformation that will reverse it and send each point back to their original location. + +```markdown +- rotate by 90º, rotate by -90º +- scale by 2, scale by 1/2 +- reflect across y-axis, reflect back across y-axis +``` + +Matrix transformations are just like the simple mathematical operators addition and multiplication, in that they can be undone by multiplying by the [inverse matrix](gloss:inverse-matrix). + +{.fixme} Explain that `A^(-1) ≠ 1/A` + +- When we multiply a number **x** by 3, we can undo that operation by multiplying by [[1/3]]. +- When we add 7 to a number **x**, we can undo that operation by adding [[-7]]. +- When we apply a transformation **A** to a vector **v**, we can undo that operation by multiplying by the [[inverse|obverse|converse|reciprocal]] of **A**. + +The **inverse matrix** of **A** is the matrix that undoes the transformation of **A**. This means When the inverse is applied, all of the points will go back to their original position. We use __`A^(-1)`__ to denote the inverse of matrix **A.** + +--- +> id: calculate-inverse + +### Example: calcuating **an inverse matrix** + +How is the inverse of a matrix calculated? Let's compare matrix multiplication to our basic operations again. + +- Multiplying **x** by 3 and 1/3 is the same as multiplying it by [[1]] (which does not change **x**). +- Given **x**, adding 7 and then subtracting 7 is the same as adding by [[0]] (which does not change **x**)**.** +- Given a vector **v**, multiplying by the matrix **A** and the inverse of that matrix **`A^(-1)`** will be the same as multiplying by [[the identity matrix|a matrix of all ones|a matrix of all zeros|one]], which does not change **v.** + + + figure: img(src="images/proto-4/Untitled.png") + + +Another way to write this is + +```markdown +**A * `A^(-1)` v = v = I v +A * `A^(-1)` = I** +``` + +**Example** + +As an example, let's look at a shear in the x-direction and calculate its inverse transformation. + +- [ ] **x-geopad:matrix([1 1, 0 1])** + +```markdown +x-algebra + +*** the inverse times the shear matrix is the identity matrix** +`A^(-1)` [1 1, 0 1] = I + +*** give our inverse variable names** +[e f, g h][1 1, 0 1] = [1 0, 0 1] + +*** expand into a system of equations** +...solve this for e,f,g,h (see Docs) +... +... + +*** answer** +`A^(-1)` = [1 -1, 0 1] + +* **confirm that we get Identity Matrix when we multiply** +[1 -1, 0 1]x[1 1, 0 1] = [1 0, 0 1] +``` + +Notice that this inverse matrix is simply a shear matrix in the opposite direction – which is what you would expect! + +- [ ] **x-geopad:** can apply transformations **A** and **`A^(-1)`** + +--- +> id: inverse-formula + +### **The general formula for an inverse matrix** + +Let's derive this formula: + + figure: image(src="images/proto-4/Screenshot_2020-11-23_at_12.16.18.png") + +More generally the equation for the inverse of a 2x2 matrix [a b, c d] is: + + figure: image(src="images/proto-4/Untitled%201.png") + +The equation for the inverse of a two-by-two matrix. + +*(we could show how multiplying the inverse x A gives the Identity Matrix)* + +What is the value in the [{.pill.red}denominator](target:denominator)? This is the [[determinant]]. + +Notice that if the determinant is zero then we cannot divide by zero. These are a specific type of matrix we will talk about later. + + figure: image(src="images/proto-4/Untitled%202.png") + +Inverse equation with determinant + +--- +> id: rotation-inverse + +### Example: Inverse of a Rotation Matrix + +In chapter 1 we derived the matrix representing a rotation through the angle *theta*. The inverse of this matrix will be the rotation through the angle [[-theta]]. Verify that a rotation matrix by -øº is the same as the inverse of a rotation matrix by øº. + +```markdown +*** Here is the rotation matrix through ø** +A = | cosø -sinø | + | sinø cosø | + +*** To find the inverse, first calculate the determinant** +det(A) = cos(ø)^2 + sin(ø)^2 +*** This is 1** +det(a) = 1 +*** So the term 1/det(A) is 1** + +* **Inverse** +`A^(-1)` = | cosø sinø | + | -sinø cosø | + +*** Think about a unit circle to see why these are true** +cos(ø) = cos(-ø) +sin(ø) = -sin(-ø) + +*** Now substitute** +`A^(-1)` = | cos(-ø) -sin(-ø) | + | sin(-ø) cos(-ø) | + +*** We have proved it!** +``` + +--- + +Verify that a rotation matrix by -x° is the same as the inverse of a rotation matrix by x°. + +{.fixme} Add some examples where students have to calculate inverses to solve problems. + +- **Possibilities** + - Ray Tracing? + - Linear Regression? + - Calculating the probability that you were in a previous step in a Markov Chain + - Linear programming, optimization? + +--- +> id: inverse-intersection + +### **Inverse matrices for solving line intersection** + +There is another way we can look at matrices. Observe the following graph of two lines with the following equations. What if we want to find the intersection of these two lines? + + figure: img(src="images/proto-4/Untitled%204.png") + + figure: img(src="images/proto-4/Untitled%205.png") + +y = x + 4 and y = 2x + 2 + +We can already observe that the lines intersect at point (2, 6). But there's also a way to solve this with matrices. How? We can rewrite these two lines as a **system of equations**, and then treat is as a matrix multiplication problem. + +```markdown +*** write the two equations** + y = x + 4 y = 2x + 2 + +*** now put variables on one side for both** +-x + y = 4 -2x + y = 2 + +*** now line up the variables** +-1*x + 1*y = 4 +-2*x + 1*y = 2 + +* **now write it as a matrix multiplication** +[ -1 1 ] | x | = | 4 | +[ -2 1 ] | y | = | 2 | + +*** we can solve for [x, y] by finding the inverse and multiplying by [4, 2]** +* **find the inverse** +1/det * [ 1 -1, 2 -1 ] + +*** find the determinant** +det = a * d - b * c = -1 - (-2) = 1 +det = 1 + +*** this is our inverse matrix** +[ 1 -1, 2 -1 ] + +*** now perform matrix multiplication against [4 2]** +[ 1 -1 ] | 4 | | x | +[ 2 -1 ] | 2 | = | y | + +*** simplify** +x = [ 4 - 2 ] = 2 +y = [ 8 - 2 ] = 6 + +*** indeed, the intersection point is...** +(2, 6) +``` + +Lesson: this was a long way to solve a problem we already knew the answer to, but it shows that we can use matrices for more than just transformations! This type of matrix calculation is very useful for higher order problems. There are better ways to solve multi-dimensional matrix problems, as we will see next chapter. + +{.fixme} Again, I think we should have 1-2 examples where students can find the intersection of two lines. + + +--- +> id: singular-matrices + +### Singular matrices (swap after next) + +The formula for the inverse of a matrix contains a determinant in the denominator. This will give us a problem when the determinant of a matrix is [[zero]]. The inverse [[cannot be calculated]]! + +Matrices with determinant zero are called **singular matrices**, meaning they do not have an inverse. + +This is just like the number 0 which has no inverse: You can multiply by 0, but you cannot divide by 0. + +Recall last chapter how the determinant of a matrix represents what happens when we apply that matrix's transformation to a set of points. If the determinant is zero, then the resulting area of the transformed space will be [[zero]]. + + figure: img(src="images/proto-4/Untitled%203.png") + +zero determinant + +{.fixme} I would maybe explain this with 1:1 or many:1 functions. A matrix transform is a 1:1 function. For every input, there is one, unique output. That makes it easy to invert it. For singular matrices, lots of points get mapped to the same output, so there is no way to tell where you came from. + +Does it make sense that a matrix with a transformation that collapses space onto zero area will not have an inverse? If the area is zero, no matter what point is put into the transformation, the resulting point will [[lie along the same line]]. + +This transformation has squeezed space onto a single line. + +There are no ways to "undo" this transformation. All of the points now lie on the same line. + +- [ ] Many-to-one, cannot go back to being a one-to-one. + +📈Geopad — like 3b1b with arrows pointing from straight line onto plane + +Here is a collection of points that have been multiplied by a **singular matrix**. Try multiplying them to get a line back out. + +📈Interactive where student can adjust a matrix that multiplies by a singular matrix, but the resulting transform will always be a line. + +--- +> id: singular-lines + +### **Non-intersecting lines (singular matrices)** + +We have seen how matrices with determinant = zero do not have inverses and are called **singular matrices**. This means that when a 2x2 matrix representing two lines is singular, it means the lines [[do not have a single solution|do not intersect|are parallel]] (tricky wording). + +**Example:** these two lines are actually the same. When we rewrite their line equations in matrix multiplication form, the matrix has a determinant of zero. There is no inverse, and thus [[no unique solution|no solution]] + + figure: img(src="images/proto-4/Untitled%206.png") + +Two lines y = -x + 3 and -4y = 4x - 12, are actually the same line + + figure: img(src="images/proto-4/Untitled%207.png") + +two equations + + figure: img(src="images/proto-4/Untitled%208.png") + +equations rewritten as a matrix + +**Example:** these two lines are parallel. When we rewrite their line equations in matrix multiplication form, it has a determinant of zero. There is no inverse, and thus [[no unique solution]] + + figure: img(src="images/proto-4/Untitled%209.png") + +Two parallel lines y=3x+2, y=3x-4, have no intersection + + figure: img(src="images/proto-4/Untitled%2010.png") + +two line equations + + figure: img(src="images/proto-4/Untitled%2011.png") + +equations rewritten as a matrix + +--- +> id: inverse-conclusion + +### Conclusion/Cliff-Hanger + +Here is the formula for calculating a 3x3 matrix. There is a way to calculate inverses of larger matrices, which we will observe next chapter. + +{.fixme} Formula for inverses of 3x3 matrices (and maybe the pattern for larger matrices) + +- Maybe a large formula? +- Maybe a list of steps that show what to do. + +"Throughout history mathematicians have done many ways to solve inverses... for example (statistics and orbit of planets). But our current methods are not enough to solve these, we need something better!" ---------------------------------------------------------------------------------------------------- diff --git a/content/matrices/images/proto-4/Screenshot_2020-11-23_at_12.16.18.png b/content/matrices/images/proto-4/Screenshot_2020-11-23_at_12.16.18.png new file mode 100755 index 0000000000000000000000000000000000000000..dd57a24cba279e5ab2c2f479775b52d0eb1e7088 GIT binary patch literal 119232 zcmeFXWprFUlQ7t3<|JljrkLAIF=GrdLz|fyVrHh8DUM@iW@ct)X6EhZedn8*vwLRt z=lxYCZ-Gkzz_idkhSoz;2w}KsW1S5OldAE zswgEYN}}jsYhrE%0sy{+C8@)ysz_mHsmI60PKu;pdSDf*7w3wj$D(qIV1A<^h5Z>$ zsH3XS_(!8MuY?S-94598@=GN{cpG!*ADR=xF4$o~DS*xc^FhbKD%VDn<3;AhczW6c zEMV_j{kO!hzkpcE(a(}FFYK6@2{;AvLa@9-go=JB=?Uc55s}DHiW18mBQIKjxcUX_ z!lw774?Xduk&!KcpO{ES_t=)Pu-7NRY7jZ*0i@7jkV}GQ6Ip0jETtm0=uh-?Y5QLU zEmHR31R+CShIy&z)KED>pqf5&0F9J8>ovDION=r?uZu{z5WGEk?>d1%I?@$MuTf2P zHK|&C;3C^z3((wmB#rG?!k9dM-}MjUJam#l3{dsKYxc}l2PM4<8(T1fDxDqyCS{o> 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zO^$oRN+0hrVIH-Vy!?LaGJt3p(=TzHmL{D$abzHTb{XAtpS6M2Dz6!tpB(qhz!k&$A#U-qJK!LsPx#*3tf>H2;Qi6#Gmjz1TrU&+6PRTIYlm0$u5b-Klv zoYW^Ah{AzH=%R2h>Dv+qV)-zKMh<#xP}`u#QGyxlovt>1COciQ_nNedkpqd)#mLMA uDi{Y6p$o>Rq*aU@NQ5p%W+qU Date: Mon, 23 Nov 2020 14:22:29 -0500 Subject: [PATCH 02/20] Quick edits to intro --- content/matrices/content.md | 35 ++++++++++------ content/matrices/functions.ts | 79 +++++++++++++++++++++++++++-------- content/matrices/styles.less | 9 ++++ 3 files changed, 93 insertions(+), 30 deletions(-) diff --git a/content/matrices/content.md b/content/matrices/content.md index 23e75b148..a25364000 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1183,7 +1183,7 @@ Let's see why this is true geometrically. > section: inverses > sectionStatus: dev -> id: inverses +> id: intro-inverses ### Intro: Inverse as the reverse of a transformation @@ -1193,21 +1193,32 @@ Every 2x2 matrix can be thought of as a linear transformation. What does it mean **Examples/Interactive:** Each of these transformations has a respective transformation that will reverse it and send each point back to their original location. -```markdown -- rotate by 90º, rotate by -90º -- scale by 2, scale by 1/2 -- reflect across y-axis, reflect back across y-axis -``` + .inverses1 + .inverse-row + .mat + x-geopad(width=150 x-axis="-3,3,1" y-axis="-3,3,1" grid padding=5): svg + .buttons + .button.transform ROTATE + .button.reverse UNDO + .mat + x-geopad(width=150 x-axis="-3,3,1" y-axis="-3,3,1" grid padding=5): svg + .buttons + .button.transform SCALE + .button.reverse UNDO + .mat + x-geopad(width=150 x-axis="-3,3,1" y-axis="-3,3,1" grid padding=5): svg + .buttons + .button.transform REFLECT + .button.reverse UNDO -Matrix transformations are just like the simple mathematical operators addition and multiplication, in that they can be undone by multiplying by the [inverse matrix](gloss:inverse-matrix). -{.fixme} Explain that `A^(-1) ≠ 1/A` +Matrix transformations are just like the simple mathematical operators addition and multiplication, in that they can be undone by multiplying by the [inverse matrix](gloss:inverse-matrix). Note that this is __not the same__ as dividing by the matrix, `1/A`. Division for matrices is not a defined operation. -- When we multiply a number **x** by 3, we can undo that operation by multiplying by [[1/3]]. -- When we add 7 to a number **x**, we can undo that operation by adding [[-7]]. -- When we apply a transformation **A** to a vector **v**, we can undo that operation by multiplying by the [[inverse|obverse|converse|reciprocal]] of **A**. +- When we multiply a number `x` by `3`, we can undo that operation by multiplying by [[1/3]]. +- When we add `7` to a number `x`, we can undo that operation by adding [[-7]]. +- When we apply a transformation `A` to a vector `v`, we can undo that operation by multiplying by the [[inverse|obverse|converse|reciprocal]] of `A`. -The **inverse matrix** of **A** is the matrix that undoes the transformation of **A**. This means When the inverse is applied, all of the points will go back to their original position. We use __`A^(-1)`__ to denote the inverse of matrix **A.** +The **inverse matrix** of matrix `A` is the matrix that undoes the transformation of `A`. This means When the inverse is applied, all of the points will go back to their original position. We use __`A^(-1)`__ to denote the inverse of matrix **A.** --- > id: calculate-inverse diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index 6aaf657f6..f21e5e42c 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -163,6 +163,24 @@ function animateTransformationOnGeo(geo: Geopad, iv: string, jv:string, m: numbe geo.animatePoint(jv, new Point(m[1][0], m[1][1]), time); } +/** + * Currently only used to draw Identity Matrix, but worth keeping. + * + * @param geo Geopad to draw on. + * @param matrix transformation to draw. + * @param name name of matrix (gets mapped to i${name} and j${name}) + */ +function drawUnitVectorsToGeo(geo: Geopad, matrix: number[][], name: string) { + geo.drawPoint('point(0,0)', + {name: 'origin', interactive: false, classes: 'hidden'}); + geo.drawPoint(`point(${matrix[0][0]},${matrix[0][1]})`, + {name: `i${name}`, interactive: false, classes: 'hidden'}); + geo.drawPoint(`point(${matrix[1][0]},${matrix[1][1]})`, + {name: `j${name}`, interactive: false, classes: 'hidden'}); + geo.drawPath(`segment(origin,i${name})`, {classes: 'red'}); + geo.drawPath(`segment(origin,j${name})`, {classes: 'green'}); +} + export function transformsCalculator($step: Step) { const $cubes = $step.$$('.cube') as ElementView[]; // they're squares, actually @@ -178,24 +196,6 @@ export function transformsCalculator($step: Step) { let c: number[][] | undefined = undefined; const I = Matrix.identity(2); - /** - * Currently only used to draw Identity Matrix, but worth keeping. - * - * @param geo Geopad to draw on. - * @param matrix transformation to draw. - * @param name name of matrix (gets mapped to i${name} and j${name}) - */ - function drawUnitVectorsToGeo(geo: Geopad, matrix: number[][], name: string) { - geo.drawPoint('point(0,0)', - {name: 'origin', interactive: false, classes: 'hidden'}); - geo.drawPoint(`point(${matrix[0][0]},${matrix[0][1]})`, - {name: `i${name}`, interactive: false, classes: 'hidden'}); - geo.drawPoint(`point(${matrix[1][0]},${matrix[1][1]})`, - {name: `j${name}`, interactive: false, classes: 'hidden'}); - geo.drawPath(`segment(origin,i${name})`, {classes: 'red'}); - geo.drawPath(`segment(origin,j${name})`, {classes: 'green'}); - } - const WAIT = 200; const ANIMATE = 500; @@ -293,3 +293,46 @@ export function determinants($step:Step) { animateTransformationOnGeo($geopad, 'ipoint', 'jpoint', [[1, 1], [-1, -1]], ANIMATE); }); } + +export function introInverses($step:Step) { + // TODO: complete + const $geopads = $step.$('.inverse-row')?.$$('x-geopad') as Geopad[]; + + const I = Matrix.identity(2); + + const names = ['rotate', 'scale', 'reflect']; + names.forEach((name, i) => { + drawUnitVectorsToGeo($geopads[i], I, name); + }); + + const $transforms = $step.$('.inverse-row')?.$$('.transform') as ElementView[]; + const matrices = [ + [[0, -1], [1, 0]], + [[2, 0], [0, 2]], + [[-1, 0], [0, 1]] + ]; + + // it just goes back to normal so we don't need these + /* const inverses = [ + [[0.5, 0], [0, 0.5]], + [[0.5, 0], [0, 0.5]], + [[0.5, 0], [0, 0.5]] + ]; */ + const $reverses = $step.$('.inverse-row')?.$$('.reverse') as ElementView[]; + + $transforms.forEach((t, i) => { + t.on('click', () => { + animateTransformationOnGeo($geopads[i], + `i${names[i]}`, `j${names[i]}`, matrices[i], 300); + }); + // $reverses[i].disable() + }); + + $reverses.forEach((t, i) => { + t.on('click', () => { + animateTransformationOnGeo($geopads[i], + `i${names[i]}`, `j${names[i]}`, I, 300); + }); + // $transforms[i].disable() + }); +} diff --git a/content/matrices/styles.less b/content/matrices/styles.less index 8a2e183b4..931fe2377 100644 --- a/content/matrices/styles.less +++ b/content/matrices/styles.less @@ -33,6 +33,15 @@ x-geopad .points circle.shot { } } +// Inverses +.inverse-row { + .mat { + display: block; + float: left; + margin: 10px; + } +} + .addition .add { display: block; float: left; From 5da5aa0655df3860eb6cb6ca041e7a7ca2e33fea Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Mon, 23 Nov 2020 15:04:21 -0500 Subject: [PATCH 03/20] Quick edits to Example Calculation --- content/matrices/content.md | 70 +++++++++++++------ content/matrices/functions.ts | 32 +++++++++ content/matrices/images/proto-4/Untitled.png | Bin 2168 -> 0 bytes content/matrices/styles.less | 4 ++ 4 files changed, 86 insertions(+), 20 deletions(-) delete mode 100755 content/matrices/images/proto-4/Untitled.png diff --git a/content/matrices/content.md b/content/matrices/content.md index a25364000..ea43384e2 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1231,47 +1231,77 @@ How is the inverse of a matrix calculated? Let's compare matrix multiplication t - Given **x**, adding 7 and then subtracting 7 is the same as adding by [[0]] (which does not change **x**)**.** - Given a vector **v**, multiplying by the matrix **A** and the inverse of that matrix **`A^(-1)`** will be the same as multiplying by [[the identity matrix|a matrix of all ones|a matrix of all zeros|one]], which does not change **v.** +{.text-center.big-equation} `A^(-1)•A=I` - figure: img(src="images/proto-4/Untitled.png") +Another way to write this is: +{.text-center.big-equation} `A^(-1)•A•v = v = I•v` -Another way to write this is +{.text-center.big-equation} `A^(-1)•A = I` -```markdown -**A * `A^(-1)` v = v = I v -A * `A^(-1)` = I** -``` -**Example** +#### Example As an example, let's look at a shear in the x-direction and calculate its inverse transformation. +{.text-center} `§[[e f] [g h]]`•`§[[1 1] [0 1]]`=`§[[1 0] [0 1]]` + - [ ] **x-geopad:matrix([1 1, 0 1])** -```markdown x-algebra -*** the inverse times the shear matrix is the identity matrix** -`A^(-1)` [1 1, 0 1] = I +{.text-center} `A^(-1)`• `§[[1 1] [0 1]]` = I + +{.caption} the inverse times the shear matrix is the identity matrix + + +{.text-center} `§[[e f] [g h]]`•`§[[1 1] [0 1]]`=`§[[1 0] [0 1]]` + +{.caption} give our inverse variable names** -*** give our inverse variable names** -[e f, g h][1 1, 0 1] = [1 0, 0 1] -*** expand into a system of equations** -...solve this for e,f,g,h (see Docs) -... -... +{.text-center} `§[[e e+f] [g g+h]]` = `§[[1 0] [0 1]]` + +{.caption} expand into a system of equations + +::: tab +#### Solve for e and f + +We know that `e = 1` and `e + f = 0` + +Sub `e = 1` into `e+f = 0` and get f = [[-1]] + +::: tab +#### Solve for g and h + +We know that `g = 0` and `g + h = 1` + +Sub `g = 0` into `g + h = 1` and get h = [[1]] + +::: + +**Answer** -*** answer** `A^(-1)` = [1 -1, 0 1] -* **confirm that we get Identity Matrix when we multiply** + + + +
[[1]][[-1]]
[[0]][[1]]
+ +--- + +{.fixme} confirm that we get Identity Matrix when we multiply [1 -1, 0 1]x[1 1, 0 1] = [1 0, 0 1] -``` Notice that this inverse matrix is simply a shear matrix in the opposite direction – which is what you would expect! -- [ ] **x-geopad:** can apply transformations **A** and **`A^(-1)`** + .inverse-row + .mat + x-geopad(width=150 x-axis="-3,3,1" y-axis="-3,3,1" grid padding=5): svg + .buttons + .button.transform A + .button.reverse INVERSE --- > id: inverse-formula diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index f21e5e42c..0d254fc6d 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -336,3 +336,35 @@ export function introInverses($step:Step) { // $transforms[i].disable() }); } + +export function calculateInverse($step: Step) { + const $geopad = $step.$('.inverse-row')?.$('x-geopad') as Geopad; + + const I = Matrix.identity(2); + + drawUnitVectorsToGeo($geopad, I, 'shear'); + + const $transform = $step.$('.inverse-row')?.$('.transform') as ElementView; + const matrix = [ + [[1, 0], [1, 1]] + ]; + + // it just goes back to normal so we don't need these + /* const inverses = [ + [[0.5, 0], [0, 0.5]], + [[0.5, 0], [0, 0.5]], + [[0.5, 0], [0, 0.5]] + ]; */ + const $reverse = $step.$('.inverse-row')?.$('.reverse') as ElementView; + + $transform.on('click', () => { + animateTransformationOnGeo($geopad, + `ishear`, `jshear`, matrix[0], 300); + }); + + $reverse.on('click', () => { + animateTransformationOnGeo($geopad, + `ishear`, `jshear`, I, 300); + }); + // $transforms[i].disable() +} diff --git a/content/matrices/images/proto-4/Untitled.png b/content/matrices/images/proto-4/Untitled.png deleted file mode 100755 index a3b93c132aa96b576f0fe521daccff43dc01b4b2..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 2168 zcmV-;2#5EHP)>S9}I0x47cv?Oe2pFP5NSpiBOCh6JRfGg~Xr&F(mRt3PwVrT)JDp z7?gmL5Z-t&YKRZgot?#ieUKt1@ZRDpon!tpd*-s;+0N`tX}g^h+Vju3eCM2h{+{z+ z&cHb598J>f>}-=-r)vHB_0Gd0R6Anw@bM&5qf*UCOchwz0Xy#nndVU#cOJ>NiNy( z`ug_g_~X|lgOBGea0Bx%9YDwI@A2TZA-SttNDUs;U^hTf_!(r-g8pNxsA?bb7_ap>fOY$2Q9$R$fj% zHU8;MF){+!`GBzry!x~InXon2)h1U6f3HVzTVaajzBtPXrg`~ z`5Yv?MS5J7b9^uaT)A!aB$@NW4uVN2e*NX#+<6d8(&OmxXYcI<|KXO^ouz!o34-aK zXU~iiQS_U8SWNdC{ph8}ES4vcVG1hTL{`hfJ)Yk>8_iYri@ej39ydA8kvpxxy=f?9 zVy~dY^@`Brl=F8i!=N@dmX6EGAi6}0+jQz@6GguoHXU~VuP z80~aX5lD~Ijxg-%eTJiaoekHDDQ$O-n;<=I$4;=*H!;^chb}jMOvb=m|2x1q{hV;4 zuYQ^v&?^;d9ZCk4J{N6Sk^am{k5h(#bJ-3t3;w{FjFO22?Eu9&i6ywi~$ca;OrzHuPPeI=9d;mC(&%!NOW^tk-lg|kpz zSy;9LN{SO>C|`VzYMKe>E4gi|sVdSnOWWWEQq`0dB_O|{Pd)k4qMkVAl4G^1nq}2T zWsxH>A&{g$?zd~+-HbmC0rCQx3V+;T1Io`|gSi#XX~(tm0;Umq2-2a1i((3xlD!3O zKVgb$oJT>6vRZqhM^Ti$Th2@RMZ0f427QjeS`s?a<32pXpw=Cs(q*EAj7HEFPCxm4 z-dD023cTG7j z?^Y#Co8>mnWfV8VZF1I<%V3Wh1pt2tDrI0o&0lk%*BpHH=2c z$|OgrY+2*PihaT%91l}M?R0irl>;LOEN3B2qY7ZaWkiITnpp{aY zToLLTdQ14@`~;2|Mpu^X#KP>!cCRHTCJH0YE)!w2!5nY-Hi)h}mm(!YWK$fF>?XTF7*N&ALEo zJ!e=3M!E_vcRZi^D85qntrDTng|Ma<<404WqiuZqILDc5%A$WUQpckKqdFuPn{<7> z)Z>(2ZZ$mmJ?&l52%D4r7N*Y#xG$tGfC$B#kFd@rw+Bk-(W_Z;KC{1Kr|vIfgwB zV`utno+^5*R>#nbi!lv_js$Mop?N%b3SARC0Db|I-{T&OoS+ZCu zO^$oRN+0hrVIH-Vy!?LaGJt3p(=TzHmL{D$abzHTb{XAtpS6M2Dz6!tpB(qhz!k&$A#U-qJK!LsPx#*3tf>H2;Qi6#Gmjz1TrU&+6PRTIYlm0$u5b-Klv zoYW^Ah{AzH=%R2h>Dv+qV)-zKMh<#xP}`u#QGyxlovt>1COciQ_nNedkpqd)#mLMA uDi{Y6p$o>Rq*aU@NQ5p%W+qU Date: Tue, 24 Nov 2020 10:04:24 -0500 Subject: [PATCH 04/20] Quick Edits to Inverse:Rotation --- content/matrices/content.md | 83 ++++++++++++------ content/matrices/functions.ts | 4 + .../matrices/images/proto-4/Untitled 1.png | Bin 2906 -> 0 bytes .../matrices/images/proto-4/Untitled 2.png | Bin 3185 -> 0 bytes 4 files changed, 58 insertions(+), 29 deletions(-) delete mode 100755 content/matrices/images/proto-4/Untitled 1.png delete mode 100755 content/matrices/images/proto-4/Untitled 2.png diff --git a/content/matrices/content.md b/content/matrices/content.md index ea43384e2..081ed1343 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1314,17 +1314,15 @@ Let's derive this formula: More generally the equation for the inverse of a 2x2 matrix [a b, c d] is: - figure: image(src="images/proto-4/Untitled%201.png") +{.text-center} `A` = `§[[a b] [c d]]` ... `A^(-1)` = `1/(ad-bc)` `§[[d (-b)] [(-c) a]]` -The equation for the inverse of a two-by-two matrix. - -*(we could show how multiplying the inverse x A gives the Identity Matrix)* +{.caption} The equation for the inverse of a two-by-two matrix. What is the value in the [{.pill.red}denominator](target:denominator)? This is the [[determinant]]. Notice that if the determinant is zero then we cannot divide by zero. These are a specific type of matrix we will talk about later. - figure: image(src="images/proto-4/Untitled%202.png") +{.text-center} `A^(-1)` = `1/det(A)` `§[[d (-b)] [(-c) a]]` Inverse equation with determinant @@ -1333,37 +1331,64 @@ Inverse equation with determinant ### Example: Inverse of a Rotation Matrix -In chapter 1 we derived the matrix representing a rotation through the angle *theta*. The inverse of this matrix will be the rotation through the angle [[-theta]]. Verify that a rotation matrix by -øº is the same as the inverse of a rotation matrix by øº. +In chapter 1 we derived the matrix representing a rotation through the angle `θ`. The inverse of this matrix will be the rotation through the angle [[-θ|180-θ]]. -```markdown -*** Here is the rotation matrix through ø** -A = | cosø -sinø | - | sinø cosø | +--- -*** To find the inverse, first calculate the determinant** -det(A) = cos(ø)^2 + sin(ø)^2 -*** This is 1** -det(a) = 1 -*** So the term 1/det(A) is 1** +::: .box.f-blue -* **Inverse** -`A^(-1)` = | cosø sinø | - | -sinø cosø | +#### Inverse of Rotation matrix -*** Think about a unit circle to see why these are true** -cos(ø) = cos(-ø) -sin(ø) = -sin(-ø) +Let's verify that the inverse of the rotation matrix by θº is the same as the rotation matrix by -θ. -*** Now substitute** -`A^(-1)` = | cos(-ø) -sin(-ø) | - | sin(-ø) cos(-ø) | +**Here is the rotation matrix through θ** -*** We have proved it!** -``` +{.text-center} A = `§[[cosθ (-sinθ)] [sinθ cosθ]]` ---- +**To find the inverse, first calculate the determinant** + +{.fixme} Formula editor here, with options cosθ, sinθ, and exponent + +{.text-center} det(A) = `cos(θ)^2 + sin(θ)^2` + +**Remembering the trigonometric identities...** -Verify that a rotation matrix by -x° is the same as the inverse of a rotation matrix by x°. +{.text-center} det(A) = [[1]] + +**The term `1/det(A)` is 1** + + +**By the formula for the Inverse matrix:** + +{.text-center} `A^(-1)` = `§[[cosθ sinθ] [(-sinθ) cosθ]]` + +**Now let's confirm this matrix is a rotation through -θ** + +**Think about a unit circle to see why these are true** + +::: column.grow + +{.text-center} `cos(θ) = cos(-θ)` + +::: column.grow + +{.text-center} `sin(θ) = -sin(-θ)` + +::: + +{.fixme} multi-choice or equation editor + +**Now substitute these values into our inverse matrix** + +{.text-center} `A^(-1)` = `§[[(cos(-θ)) (-sin(-θ))] [(sin(-θ)) (cos(-θ))]]` + +**This is the same as the rotation matrix for -θ.** + +**We have proved it!** + +::: + +--- {.fixme} Add some examples where students have to calculate inverses to solve problems. @@ -1434,7 +1459,7 @@ Lesson: this was a long way to solve a problem we already knew the answer to, bu --- > id: singular-matrices -### Singular matrices (swap after next) +### Singular matrices The formula for the inverse of a matrix contains a determinant in the denominator. This will give us a problem when the determinant of a matrix is [[zero]]. 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Observe the following graph of two lines with the following equations. What if we want to find the intersection of these two lines? - figure: img(src="images/proto-4/Untitled%204.png") + x-geopad(width=300 x-axis="-8,8,1" y-axis="-8,8,1" grid axes): svg + path.blue(x="line(point(0,4),point(2,6))" target="l1") + path.green(x="line(point(0,2),point(2,6))" target="l2") - figure: img(src="images/proto-4/Untitled%205.png") - -y = x + 4 and y = 2x + 2 +[{.pill.blue}y = x + 4](target:l1) and [{.pill.green}y = 2x + 2](target:l2) We can already observe that the lines intersect at point (2, 6). But there's also a way to solve this with matrices. How? We can rewrite these two lines as a **system of equations**, and then treat is as a matrix multiplication problem. -```markdown -*** write the two equations** - y = x + 4 y = 2x + 2 +::: .box.f-yellow + +#### Intersection + +**Write the two equations** + +::: column.grow +{.text-center} [{.pill.blue}`y = x + 4`](target:l1) +::: column.grow +{.text-center} [{.pill.green}`y = 2x + 2`](target:l2) +::: + +**For both equations, isolate the variables on one side.** + +::: column.grow +{.text-center} [{.pill.blue}`-x + y = 4`](target:l1) +::: column.grow +{.text-center} [{.pill.green}`-2x + y = 2`](target:l2) +::: + +**Line up the variables and write the coefficients** -*** now put variables on one side for both** --x + y = 4 -2x + y = 2 +{.text-center} [{.pill.blue}`-1*x + 1*y = 4`](target:l1) -*** now line up the variables** --1*x + 1*y = 4 --2*x + 1*y = 2 +{.text-center} [{.pill.green}`-2*x + 1*y = 2`](target:l2) -* **now write it as a matrix multiplication** -[ -1 1 ] | x | = | 4 | -[ -2 1 ] | y | = | 2 | +**Rewrite it as a matrix multiplication** -*** we can solve for [x, y] by finding the inverse and multiplying by [4, 2]** -* **find the inverse** -1/det * [ 1 -1, 2 -1 ] +{.text-center} `§[[(-1) 1] [(-2) 1]]` • `§[[x] [y]]` = `§[[4] [2]]` -*** find the determinant** -det = a * d - b * c = -1 - (-2) = 1 -det = 1 -*** this is our inverse matrix** -[ 1 -1, 2 -1 ] +**We can solve for [x, y] by finding the inverse and multiplying by [4, 2]** -*** now perform matrix multiplication against [4 2]** -[ 1 -1 ] | 4 | | x | -[ 2 -1 ] | 2 | = | y | +**Write the formula for the inverse** -*** simplify** -x = [ 4 - 2 ] = 2 -y = [ 8 - 2 ] = 6 +{.text-center} `1/det(A)` • `§[[1 (-1)] [2 (-1)]]` + +**Find the determinant** + +{.text-center} `det(A)` = `a * d - b * c` = `-1 - (-2)` = `1` + +{.text-center} `det(A) = 1` + +**This is our inverse matrix** + +{.text-center} `A^(-1)` = `§[[1 (-1)] [2 (-1)]]` + +**Now perform matrix multiplication with the inverse and [4 2]** + +{.text-center} `§[[1 (-1)] [2 (-1)]]` • `§[[4] [2]]` = `§[[x] [y]]` + +**Simplify** +::: column.grow +{.text-center} `x = [ 4 - 2 ] = 2` +::: column.grow +{.text-center} `y = [ 8 - 2 ] = 6` +::: *** indeed, the intersection point is...** (2, 6) -``` + +::: Lesson: this was a long way to solve a problem we already knew the answer to, but it shows that we can use matrices for more than just transformations! This type of matrix calculation is very useful for higher order problems. There are better ways to solve multi-dimensional matrix problems, as we will see next chapter. From 1a5cd72a0047b6ad9395056c2d4eba8132a1e72e Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Tue, 24 Nov 2020 11:56:15 -0500 Subject: [PATCH 06/20] Quick Edits to Inverse:Derive --- content/matrices/content.md | 76 +++++++++++++++++- .../Screenshot_2020-11-23_at_12.16.18.png | Bin 119232 -> 0 bytes .../matrices/images/proto-4/Untitled 4.png | Bin 36929 -> 0 bytes .../matrices/images/proto-4/Untitled 5.png | Bin 53730 -> 0 bytes 4 files changed, 72 insertions(+), 4 deletions(-) delete mode 100755 content/matrices/images/proto-4/Screenshot_2020-11-23_at_12.16.18.png delete mode 100755 content/matrices/images/proto-4/Untitled 4.png delete mode 100755 content/matrices/images/proto-4/Untitled 5.png diff --git a/content/matrices/content.md b/content/matrices/content.md index 530ab8a60..d282db68e 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1306,15 +1306,81 @@ Notice that this inverse matrix is simply a shear matrix in the opposite directi --- > id: inverse-formula -### **The general formula for an inverse matrix** +### The general formula for an inverse matrix + + +::: .box.f-green + +#### Deriving the Formula for the Inverse Let's derive this formula: - figure: image(src="images/proto-4/Screenshot_2020-11-23_at_12.16.18.png") +**We want to solve for this** + +{.text-center} `§[[a b] [c d]]` • `§[[w x] [y z]]` = `§[[1 0] [0 1]]` + +**Write out the meaning** + +::: column.grow +{.text-center} __eq1__ `aw + by = 1` + +{.text-center} __eq3__ `cw + dy = 0` +::: column.grow +{.text-center} __eq2__ `ax + bz = 0` + +{.text-center} __eq4__ `cx + dz = 1` +::: + + +**Solve for y** + +{.text-center} Multiply-subtract `c`•__eq1__ - `a`•__eq3__ + +{.text-center} `caw + cby = c` + +{.text-center} `caw + ady = 0` + +{.text-center} `0 + (cb - ad)y = c` + +{.text-center} `y = c/(cb-ad)` + +**Solve for w** -More generally the equation for the inverse of a 2x2 matrix [a b, c d] is: +{.text-center} Substitute `y` into __eq3__ -{.text-center} `A` = `§[[a b] [c d]]` ... `A^(-1)` = `1/(ad-bc)` `§[[d (-b)] [(-c) a]]` +{.text-center} `cw + dc/(cb-ad) = 0` + +{.text-center} `w = -d/(cb-ad)` + +**Solve for x** + +{.text-center} Multiply-add `d`•__eq2__ - `b`•__eq4__ + +{.text-center} `d`•__eq2__ --> `dax + dbz = 0` + +{.text-center} `b`•__eq4__ --> `bcx + bdz = b` + +{.text-center} `(da - bc)x + 0 = -b` + +{.text-center} `x = -b/(ad - bc)` + +**Solve for z** + +{.text-center} Substitute `x` into __eq2__ + +{.text-center} `a(-b/(ad-bc)) + bz = 0` + +{.text-center} `-a/(ad-bc) + z = 0` + +{.text-center} `z = a/(ad-bc)` + +**Put them all together** + +{.text-center} `§[[(d/(ad-bc)) (-b/(ad-bc))] [(-c/(ad-bc)) (a/(ad-bc))]]` + +**Extract the common factor** + +{.text-center} `1/(ad-bc)` • `§[[d (-b)] [(-c) a]]` {.caption} The equation for the inverse of a two-by-two matrix. @@ -1326,6 +1392,8 @@ Notice that if the determinant is zero then we cannot divide by zero. 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bb0c70b5a6f457491ff4b9a0b79376d0ce9ef356 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Tue, 24 Nov 2020 17:52:40 -0500 Subject: [PATCH 07/20] Migrate part of Gauss draft from Notion into Markdown --- content/matrices/content.md | 137 ++++++++++++++++++++++++++++++++++++ 1 file changed, 137 insertions(+) diff --git a/content/matrices/content.md b/content/matrices/content.md index d282db68e..c9dc5e8f4 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1643,6 +1643,143 @@ Here is the formula for calculating a 3x3 matrix. There is a way to calculate in {.todo} COMING SOON +> id: two-equations + +::: column.grow + +**write the two equations** +```markdown +EQ1: y = x + 4 +EQ2: y = 2x + 2 +``` +EQ1: [{.pill.blue}y = x + 4](target:l1) +EQ2: [{.pill.green}y = 2x + 2](target:l2) + +**now put variables on the same side** +```markdown +EQ1: -x + y = 4 +EQ2: -2x + y = 2 +``` +EQ1: [{.pill.blue}-x + y = 4](target:l1) +EQ2: [{.pill.green}-2x + y = 2](target:l2) + +**now write in augmented matrix form** +```markdown +R1 | -1 1 : 4 | +R2 | -2 1 : 2 | +``` + +::: column(width=200) + +**original system of equations** + + x-geopad(width=300 x-axis="-8,8,1" y-axis="-8,8,1" grid axes): svg + path.blue(x="line(point(0,4),point(2,6))" target="l1") + path.green(x="line(point(0,2),point(2,6))" target="l2") + +{.caption} y = x + 4; y = 2x + 2; + +::: + +[Continue](btn:next) + +--- + +::: column.grow + +**now start row-reduction** + +**first goal: 0s in bottom left** +```markdown +R1 | -1 1 : 4 | +1/2*R2 -> R2 | -1 1/2 : 1 | + +R1 | -1 1 : 4 | +R2-R1 -> R2 | 0 -1/2 : -3 | + +``` + +**simplify (make coefficient a 1)** +```markdown +R1 | -1 1 : 4 | +-2*R2 -> R2 | 0 1 : 6 | +``` + +::: column(width=200) + +**after R2 - R1 → R2** + + x-geopad(width=300 x-axis="-8,8,1" y-axis="-8,8,1" grid axes): svg + path.blue.dashed(x="line(point(0,4),point(2,6))" target="l1") + path.blue(x="line(point(0,6),point(2,6))" target="l1p") + path.green(x="line(point(0,2),point(2,6))" target="l2") + +{.caption} y = x + 4; y = 6; + +{.fixme} animate this as row is substituted! + +::: + +[Continue](btn:next) + +--- + +::: column.grow + +**now perform back-substitution** +```markdown +R1-R2 -> R1 | -1 0 : -2 | +R2 | 0 1 : 6 | +``` + +**and simplify** +```markdown +-1*R1 -> R1 | 1 0 : 2 | +R2 | 0 1 : 6 | +``` + +::: column(width=200) + +**after back-substitution** + +**R1 - R2 → R1** + + x-geopad(width=300 x-axis="-8,8,1" y-axis="-8,8,1" grid axes): svg + path.blue(x="line(point(0,6),point(2,6))" target="l1p") + path.green.dashed(x="line(point(0,2),point(2,6))" target="l2") + path.green(x="line(point(2,2),point(2,6))" target="l2p") + +{.caption} x = 2; y = 6; + +{.fixme} animate this as row is substituted! + +::: + +[Continue](btn:next) + +--- + +::: column.grow + +**we have our solution!** +```markdown +x = 2 +y = 6 +``` +[{.pill.green}x = 2](target:l2p) +[{.pill.blue}y = 6](target:l1p) + +::: column(width=200) + + x-geopad(width=300 x-axis="-8,8,1" y-axis="-8,8,1" grid axes): svg + path.blue(x="line(point(0,6),point(2,6))" target="l1p") + path.green(x="line(point(2,2),point(2,6))" target="l2p") + +{.caption} x = 2; y = 6; + +::: + + ---------------------------------------------------------------------------------------------------- From d0a10436e280d2a298bfb07bf418a9eb66f849e4 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Tue, 24 Nov 2020 21:52:00 -0500 Subject: [PATCH 08/20] Prototype for Network Analysis --- content/matrices/content.md | 37 ++++++++++++++++++ .../matrices/images/proto-5/network-label.gif | Bin 0 -> 490084 bytes 2 files changed, 37 insertions(+) create mode 100644 content/matrices/images/proto-5/network-label.gif diff --git a/content/matrices/content.md b/content/matrices/content.md index c9dc5e8f4..3440ca21f 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1779,6 +1779,43 @@ y = 6 ::: +--- + +> id: network-analysis + +::: .box.f-blue +#### Network analysis + +::: column.grow + +**Write a system of equations representing this network.** + +```markdown +e1 + e7 - e5 - e6 = 0 +-e1 + e2 = 0 +e3 - e2 - e7 = 0 +-e3 + e4 + e6 = 0 +-e4 + e5 = 0 +``` + +**Now write it as an augmented matrix** +```markdown +| 1 0 0 0 -1 -1 1 : 0 | +| -1 1 0 0 0 0 0 : 0 | +| 0 -1 1 0 0 0 -1 : 0 | +| 0 0 -1 1 0 1 0 : 0 | +| 0 0 0 -1 1 0 0 : 0 | +``` + +::: column(width=300) + // prototype here: + // https://editor.p5js.org/kevindeland/sketches/ud4f-fl-b + img(src="images/proto-5/network-label.gif") +::: + +::: + +--- 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zXwV}*F^H!!z$bOpQ{L2T;v3jq{7g8x7uk?C&R1<}kis_^B}P^y4jh9M2DbkJH${%i literal 0 HcmV?d00001 From e8d8a66537b5a2ed2bdbaec6d95feb4bbbd8d613 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Wed, 25 Nov 2020 14:54:06 -0500 Subject: [PATCH 09/20] Draft for Inverse:LeastSquares --- content/matrices/content.md | 84 +++++++++++++++++++++++++++++++++++++ 1 file changed, 84 insertions(+) diff --git a/content/matrices/content.md b/content/matrices/content.md index 3440ca21f..cc4d418d2 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1617,6 +1617,90 @@ two line equations equations rewritten as a matrix +--- +> id: least-squares + +The "Method of Least Squares" is a way to estimate a trend for data. + +::: .box.f-yellow +#### Method of Least Squares + +{.fixme} Different points (these are copied from a LinAlg textbook) + +We have a set of points. Can we find a best fit line? + +::: column.grow + + - points = [[1,1], [2,2], [3,4], [4,4], [5,6]] + table.data + tr: td var1 + td var2 + for p in points + tr + td= p[0] + td= p[1] + +::: column(width=300) + + - points = [[1,1], [2,2], [3,4], [4,4], [5,6]] + x-geopad(width=300 x-axis="-1, 8, 1" y-axis="-1, 8, 1" grid axes): svg + for p,i in points + circle.blue(name=`p${i}` x=`point(${p[0]}, ${p[1]})`) + path.red(x=`segment(point(${p[0]},${p[1]}), point(${p[0]},${1*p[0]+0}))` target="error") + path.blue(x="line(point(0,0),point(1,1))" target="guess1") + +::: + +We want to see how a best-fit line would work. + +{.text-center} `y = mx + b` + +We want to find some function `f(x)` that fits this line best. A possible best fit line is [{.pill.blue}y = x](target:guess1). + +How can we minimize the total [{.pill.red}error](target:error)? + +{.fixme} is it possible to automate this, i.e. put it in PUG? + +::: column.grow +**Write X like this** + +`X` = `§[[1 1] [1 2] [1 3] [1 4] [1 5]]` +::: column.grow +**Write Y like this** + +`Y` = `§[[1] [2] [4] [4] [6]]` +::: + +{.text-center} `Y = XA + E` + +**The formula to minimize the error** + +{.text-center} `A = (X^(T)*X)^(-1)*X^T*Y` + +**New Notation** + +`X^T` is the [transpose](gloss:transpose-matrix) of the matrix `X`. This is a very simple operation, it just means you switch the rows and the columns. + +```markdown +X = [...]5x2 +X^T = [...]2x5 + +X^(T)•X = [....]2x2 +(X^(T)•X)^-1 = 1/det[...]2x2 + +**now multiply it by Y** +NY = [...]2x1 + +**and write the equation** +f(x): y = mx + b + +// here is where we draw the line +``` +WOW! +this is truly magic. To learn more about how this works, check the course on statistical methods and machine learning + +::: + --- > id: inverse-conclusion From 801fbb003164d046224202e2f04d68abfab72990 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Wed, 25 Nov 2020 15:41:50 -0500 Subject: [PATCH 10/20] Draft for Gauss:ElectricBill --- content/matrices/content.md | 53 ++++++++++++++++++++++++++++++++++++- 1 file changed, 52 insertions(+), 1 deletion(-) diff --git a/content/matrices/content.md b/content/matrices/content.md index cc4d418d2..bbf05c344 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1868,7 +1868,9 @@ y = 6 > id: network-analysis ::: .box.f-blue -#### Network analysis +#### 🚦 Traffic Network Analysis + +[Networks](linkto:networks) are everywhere. We find networks in economic models, electric circuits, and traffic patterns. ::: column.grow @@ -1899,6 +1901,55 @@ e3 - e2 - e7 = 0 ::: +--- +> id: electric-bill + +::: .box.f-blue +#### 🔌 Electric Bill + +A restaurant owner wants to measure the power consumption of her most commonly used appliances. + +The three main uses of power are: + +🥶 Freezer +🍕 Oven +🧼 Dishwasher + +She wants to calculate how much it costs to run each appliance for an hour. + +She has three weekly electric bills and also knows how often she was using each appliance for that week. + +**Hours + Bills per Week** + +| Week | Freezer | Oven | Dishwasher | Bill | +|---|---|---|---|---| +| Week 1 | 168 | 20 | 20 | $1400 | +| Week 2 | 168 | 28 | 12 | $1336 | +| Week 3 | 168 | 42 | 18 | $1824 | + + + // use values: f=5, o=20, d=8 + - x = [5, 20, 8] + - A = [[168, 20, 20], [168, 20, 12], [168, 42, 18]] + - w1 = A[0][0]*x[0] + A[0][1]*x[1] + A[0][2]*x[2] + - w2 = A[1][0]*x[0] + A[1][1]*x[1] + A[1][2]*x[2] + - w3 = A[2][0]*x[0] + A[2][1]*x[1] + A[2][2]*x[2] + // ul Answers: + // li= w1 + // li= w2 + // li= w3 + +**Rewrite this as a matrix** + +{.text-center} `Cost` = `A` * `hourly` + +{.text-center} `§[[1400] [1336] [1824]]` = `§[[168 20 20] [168 28 12] [168 42 18]]` • `§[[f] [o] [d]]` + +**Now perform Gaussian Elimination to find the cost-per-hour of each appliance** + +::: + + --- ---------------------------------------------------------------------------------------------------- From 1bc5cb091ca5756c3d50111c6f76913e1c4286b1 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Thu, 3 Dec 2020 11:16:17 -0500 Subject: [PATCH 11/20] Beginnings of x-gaussian solver --- content/matrices/components/gaussian.pug | 8 ++ content/matrices/components/gaussian.ts | 140 +++++++++++++++++++++++ content/matrices/content.md | 31 +++++ content/matrices/functions.ts | 6 + content/matrices/styles.less | 14 +++ 5 files changed, 199 insertions(+) create mode 100644 content/matrices/components/gaussian.pug create mode 100644 content/matrices/components/gaussian.ts diff --git a/content/matrices/components/gaussian.pug b/content/matrices/components/gaussian.pug new file mode 100644 index 000000000..aedfabc46 --- /dev/null +++ b/content/matrices/components/gaussian.pug @@ -0,0 +1,8 @@ +.gaussian + svg.equation: g + .matrix: slot(name="matrix") + .rows: slot(name="rows") + .nav + // .btn.back.hide: +icon('left', 24) + // .btn.next: +icon('right', 24) + .legend-box: slot diff --git a/content/matrices/components/gaussian.ts b/content/matrices/components/gaussian.ts new file mode 100644 index 000000000..db8345cab --- /dev/null +++ b/content/matrices/components/gaussian.ts @@ -0,0 +1,140 @@ +import {$N, CustomElementView, ElementView, register, SVGParentView, SVGView} from '@mathigon/boost'; +import template from './gaussian.pug'; + +// for setting +const FRAME_WIDTH = 600; +const FRAME_HEIGHT = 200; + +@register('x-gaussian', {template, attributes: ['dimension', 'matrix']}) +export class Gaussian extends CustomElementView { + + private originalValues: number[][]; + private $frame!: SVGParentView; + private $matrices!: SVGParentView[]; + + private $leftMatrix!: SVGView; + private $row1!: SVGView; + private $row2!: SVGView; + + private $row3?: SVGView; + + private numVars!: number; + private numRows!: number; + private vars!: string[]; + + private A!: number[][]; + private c!: number[]; + private rows!: number[][]; + + private $rows!: SVGView[]; + + ready() { + console.log('This is the Gaussian'); + // this.originalValues = this.attr('matrix'); + // this.originalValues = [[-1, 1, 4], [-2, 1, 2]]; + + this.$frame = $N('svg', {viewBox: `0 0 ${FRAME_WIDTH} ${FRAME_HEIGHT}`}, this) as SVGParentView; + // this.$matrices = []; + + this.parseMatrix(); + this.$('.matrix')?.hide(); + + this.displayRows(); + } + + parseMatrix() { + const matrices = this.$('.matrix')?.$$('x-math') as ElementView[]; + if (matrices.length != 3) throw new Error('Three matrices expected'); + + const stripParens = (str: string) => str.replace(/\(|\)|\s/g, ''); + const replaceMinus = (str: string) => str.replace(/−/g, '-'); + + // Matrix 1: A + const numbers1 = matrices[0].$('desc')?.text.split(',') + .map(stripParens) + .map(replaceMinus) // different minus sign? + .map(str => parseInt(str)) as number[]; + + // Matrix 2: var names + const vars2 = matrices[1].$('desc')?.text.split(',') + .map(stripParens) as string[]; + this.numVars = vars2.length; + this.vars = vars2.map(v => v); + + // Matrix 3: solution + const numbers3 = matrices[2].$('desc')?.text.split(',') + .map(stripParens) + .map(replaceMinus) // different minus sign? + .map(str => parseInt(str)) as number[]; + + this.A = []; + this.rows = []; + this.c = numbers3; + this.numRows = this.c?.length; + for (let r = 0; r < this.numRows!; r++) { + const Arow = numbers1.slice(r * this.numVars!, r * this.numVars! + this.numVars!); + this.A.push(Arow); + this.rows.push(Arow.concat(this.c[r])); + } + + console.log(this.A); + console.log(this.c); + console.log(this.rows); + } + + displayRow($row: SVGView, rowNumber: number) { + + const allTextAttr = { + class: 'row-text', + 'dominant-baseline': 'central', + fill: 'white' + }; + + const y = 14 + rowNumber * 50; + const fillAttr = { + class: 'row-fill', x: 13, y, width: this.numVars === 2 ? 160 : 240, height: 37, rx: 4 + }; + + const divX = this.numVars === 2 ? 129 : 175; + const dividerAttr = { + x1: divX, x2: divX, stroke: 'white', 'stroke-width': '2', 'stroke-dasharray': '2 2' + }; + + const _rowFill = $N('rect', Object.assign({y}, fillAttr), $row) as SVGView; + const textAttr = Object.assign({y: y + 37 / 2}, allTextAttr); + const _divider = $N('line', Object.assign({y1: y, y2: y + 37}, dividerAttr), $row); + + + const _text1 = $N('text', Object.assign({x: 20, text: this.rows[rowNumber][0]}, textAttr), $row) as SVGView; + const _text2 = $N('text', Object.assign({x: 74, text: this.rows[rowNumber][1]}, textAttr), $row) as SVGView; + + if (this.numVars === 2) { + const _text3 = $N('text', Object.assign({x: 134, text: this.rows[rowNumber][2]}, textAttr), $row) as SVGView; + } else if (this.numVars === 3) { + const _text3 = $N('text', Object.assign({x: (74 + 74 - 20), text: this.rows[rowNumber][2]}, textAttr), $row) as SVGView; + const _text4 = $N('text', Object.assign({x: (74 + 74 - 20 + 134 - 74), text: this.rows[rowNumber][3]}, textAttr), $row) as SVGView; + } + + } + + displayRows() { + this.$rows = []; + this.$leftMatrix = $N('g', {x: 30, y: 25}, this.$frame) as SVGView; + + // FIXME: size should be dependent on numbers of rows (numRows) and columns (numVars) + const matrixHeight = this.numRows * 37 + (this.numRows + 1) * 13; + const fillAttr = { + class: 'matrix', x: 0, y: 0, width: 186, height: matrixHeight, rx: 0 + }; + + const _fill = $N('rect', fillAttr, this.$leftMatrix) as SVGView; + + const rowAttr = {class: 'row-container', x: 13}; + for (let row = 0; row < this.numRows; row++) { + const y = 14 + row * 50; + const $row = $N('g', Object.assign({y}, rowAttr), this.$leftMatrix) as SVGView; + this.displayRow($row, row); + this.$rows.push($row); + } + } +} diff --git a/content/matrices/content.md b/content/matrices/content.md index bbf05c344..ba16ec63a 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1725,8 +1725,39 @@ Here is the formula for calculating a 3x3 matrix. There is a way to calculate in > sectionStatus: dev {.todo} COMING SOON +> id: gaussian-solve +Here is a Gaussian component +::: x-gaussian(dimension="2x2") + +{div(slot="matrix")} `§[[(-1) 1] [(-2) 1]]` • `§[[x] [y]]` = `§[[4] [2]]` + +* Here is step one +* Here is step two +* Here is step three + +::: + +--- +> id: gaussian-solve2 + +::: x-gaussian(dimension="3x3") + +{div(slot="matrix")} `§[[168 20 20] [168 28 12] [168 42 18]]` • `§[[x] [y] [z]]` = `§[[1400] [1336] [1824]]` + +::: + +--- +> id: gaussian-solve3 + +::: x-gaussian(dimension="5x7") + +{div(slot="matrix")} `§[[1 0 0 0 -1 -1 1] [-1 1 0 0 0 0 0] [0 -1 1 0 0 0 -1] [0 0 -1 1 0 1 0] [0 0 0 -1 1 0 0]]` • `§[[e1] [e2] [e3] [e4] [e5] [e6] [e7]]` = `§[[0] [0] [0] [0] [0]]` + +::: + +--- > id: two-equations ::: column.grow diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index 1841678d2..15bc948f1 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -11,6 +11,8 @@ import {Angle, ORIGIN, Point} from '@mathigon/euclid'; import {ElementView, ScreenEvent, svgPointerPosn} from '@mathigon/boost'; import {Geopad, Step} from '../shared/types'; +import './components/gaussian'; + export function rocket($step: Step) { const $geo = $step.$('x-geopad') as Geopad; @@ -372,3 +374,7 @@ export function calculateInverse($step: Step) { export function rotationInverse(_$step: Step) { // TODO: put an x-matrix-solver here } + +export function gaussianSolve(_$step: Step) { + // okay? +} diff --git a/content/matrices/styles.less b/content/matrices/styles.less index 7a95d2730..1824575a4 100644 --- a/content/matrices/styles.less +++ b/content/matrices/styles.less @@ -116,4 +116,18 @@ table { &.blue x-var { background: @blue; } +} + + +// for Gaussian +rect.matrix { + fill: #C3E0FC +} +rect.row-fill { + fill: @blue; +} + +text.row-text { + fill: '#FFFFFF'; + font-size: 20; } \ No newline at end of file From 27666b9531c21d0edbaa831e280e6101ccf5cdd7 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Thu, 3 Dec 2020 12:40:10 -0500 Subject: [PATCH 12/20] Can copy rows, can move left and right matrices --- content/matrices/components/gaussian.pug | 9 ++- content/matrices/components/gaussian.ts | 90 +++++++++++++++++++++--- content/matrices/components/left.svg | 1 + content/matrices/components/right.svg | 1 + content/matrices/functions.ts | 24 ++++++- content/matrices/styles.less | 3 + 6 files changed, 117 insertions(+), 11 deletions(-) create mode 100644 content/matrices/components/left.svg create mode 100644 content/matrices/components/right.svg diff --git a/content/matrices/components/gaussian.pug b/content/matrices/components/gaussian.pug index aedfabc46..0f76c3bd5 100644 --- a/content/matrices/components/gaussian.pug +++ b/content/matrices/components/gaussian.pug @@ -1,8 +1,13 @@ +mixin icon(url, width, height) + - height_ = height || width + svg(viewBox=`0 0 ${width} ${height_}` class="icon" width=width height=height_)&attributes(attributes) + use(xlink:href=`${url}.svg`) + .gaussian svg.equation: g .matrix: slot(name="matrix") .rows: slot(name="rows") .nav - // .btn.back.hide: +icon('left', 24) - // .btn.next: +icon('right', 24) + .btn.back.hide: +icon('left', 24) + .btn.next: +icon('right', 24) .legend-box: slot diff --git a/content/matrices/components/gaussian.ts b/content/matrices/components/gaussian.ts index db8345cab..5ed55a77f 100644 --- a/content/matrices/components/gaussian.ts +++ b/content/matrices/components/gaussian.ts @@ -13,10 +13,7 @@ export class Gaussian extends CustomElementView { private $matrices!: SVGParentView[]; private $leftMatrix!: SVGView; - private $row1!: SVGView; - private $row2!: SVGView; - - private $row3?: SVGView; + private $rightMatrix!: SVGView; private numVars!: number; private numRows!: number; @@ -28,6 +25,8 @@ export class Gaussian extends CustomElementView { private $rows!: SVGView[]; + private stepNumber = 0; + ready() { console.log('This is the Gaussian'); // this.originalValues = this.attr('matrix'); @@ -42,7 +41,29 @@ export class Gaussian extends CustomElementView { this.displayRows(); } - parseMatrix() { + nextStep() { + console.log(`On Step ${this.stepNumber}`); + const steps = [ + () => this.createBlankRightMatrix(), // 0 + () => this.drawArrow(), // 1 + () => this.copyRow(0), // 2 + () => this.copyRow(1), // 3 + () => this.moveRightMatrixToLeft(), // 4 + + () => this.createBlankRightMatrix(), // 5 + () => this.copyRow(0), + () => this.copyRow(1), + () => this.moveRightMatrixToLeft() + ]; + steps[this.stepNumber](); + this.stepNumber++; // = this.stepNumber + 1; + } + + backStep() { + // TODO: + } + + private parseMatrix() { const matrices = this.$('.matrix')?.$$('x-math') as ElementView[]; if (matrices.length != 3) throw new Error('Three matrices expected'); @@ -82,7 +103,7 @@ export class Gaussian extends CustomElementView { console.log(this.rows); } - displayRow($row: SVGView, rowNumber: number) { + private displayRow($row: SVGView, rowNumber: number) { const allTextAttr = { class: 'row-text', @@ -114,10 +135,9 @@ export class Gaussian extends CustomElementView { const _text3 = $N('text', Object.assign({x: (74 + 74 - 20), text: this.rows[rowNumber][2]}, textAttr), $row) as SVGView; const _text4 = $N('text', Object.assign({x: (74 + 74 - 20 + 134 - 74), text: this.rows[rowNumber][3]}, textAttr), $row) as SVGView; } - } - displayRows() { + private displayRows() { this.$rows = []; this.$leftMatrix = $N('g', {x: 30, y: 25}, this.$frame) as SVGView; @@ -137,4 +157,58 @@ export class Gaussian extends CustomElementView { this.$rows.push($row); } } + + createBlankRightMatrix() { + console.log('createBlankRightMatrix'); + this.$rightMatrix = $N('g', {x: 400, y: 25}, this.$frame) as SVGView; + console.log(this.$rightMatrix); + const matrixHeight = this.numRows * 37 + (this.numRows + 1) * 13; + const fillAttr = { + class: 'matrix-right', x: 400, y: 0, width: 186, height: matrixHeight, rx: 0 + }; + const _fill = $N('rect', fillAttr, this.$rightMatrix) as SVGView; + } + + drawArrow() { + // TODO: check fibonacci for how arrow/paths are drawn + } + + copyRow(_rowNumber: number) { + const newRow = this.$rows[_rowNumber].copy(true, false); + this.$rightMatrix.append(newRow); + newRow.animate({transform: [ + `translate(0px, 0px)`, + `translate(400px, 0px)` + ]}, 800, 500); + } + + multiplyRow(_rowNumber: number, _multiplier: number) { + // TODO: draw arrow with a multiplier circle + } + + comboRow(_rowSrc1: number, _rowSrc2: number, _rowDest: number, _m1: number, _m2: number) { + // TODO: draw multiple arrows into one + } + + moveRightMatrixToLeft() { + // remove old Left. + this.$leftMatrix.remove(); + // copy and append to frame + this.$leftMatrix = this.$rightMatrix.copy(true, false) as SVGView; + this.$frame.append(this.$leftMatrix); + // change fill to indicate it's our current matrix + const fill = this.$leftMatrix.$('.matrix-right'); + fill?.setClass('matrix-right', false); + fill?.setClass('matrix', true); + // remove the right matrix + this.$rightMatrix.remove(); + // animate it to the left + this.$leftMatrix.animate({ + transform: [ + `translate(0px, 0px)`, + `translate(-400px, 0px)` + ] + }, 800, 0); + } + } diff --git a/content/matrices/components/left.svg b/content/matrices/components/left.svg new file mode 100644 index 000000000..9ceccd539 --- /dev/null +++ b/content/matrices/components/left.svg @@ -0,0 +1 @@ + \ No newline at end of file diff --git a/content/matrices/components/right.svg b/content/matrices/components/right.svg new file mode 100644 index 000000000..46c1152b5 --- /dev/null +++ b/content/matrices/components/right.svg @@ -0,0 +1 @@ + \ No newline at end of file diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index 15bc948f1..a734db7df 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -12,6 +12,7 @@ import {ElementView, ScreenEvent, svgPointerPosn} from '@mathigon/boost'; import {Geopad, Step} from '../shared/types'; import './components/gaussian'; +import {Gaussian} from './components/gaussian'; export function rocket($step: Step) { @@ -376,5 +377,26 @@ export function rotationInverse(_$step: Step) { } export function gaussianSolve(_$step: Step) { - // okay? + const $gauss = _$step.$('x-gaussian') as Gaussian; + + const back = $gauss.$('.btn.back'); + const next = $gauss.$('.btn.next'); + // FIXME: create a "step-setter" function for Gaussians + // FIXME: possibly copy the cheesecake thing in how the next steps work + // $gauss.setSteps({}); + + back?.on('click', () => { + $gauss.backStep(); + }); + + next?.on('click', () => { + $gauss.nextStep(); + }); + + // TODO: pseudocode + const steps = [ + 'new', + 'r1.r1', + 'r2=0.5*r2' + ]; } diff --git a/content/matrices/styles.less b/content/matrices/styles.less index 1824575a4..07843d967 100644 --- a/content/matrices/styles.less +++ b/content/matrices/styles.less @@ -123,6 +123,9 @@ table { rect.matrix { fill: #C3E0FC } +rect.matrix-right { + fill: #C8EAC8 +} rect.row-fill { fill: @blue; } From 11d27d8506ae6d8493d0f4f6d04ebc04df3b477e Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Thu, 3 Dec 2020 15:32:03 -0500 Subject: [PATCH 13/20] Fully articulated and half-implemented steps for solving 2x2 --- content/matrices/components/gaussian.ts | 99 ++++++++++++++++++++++--- content/matrices/styles.less | 5 ++ 2 files changed, 94 insertions(+), 10 deletions(-) diff --git a/content/matrices/components/gaussian.ts b/content/matrices/components/gaussian.ts index 5ed55a77f..acd24415c 100644 --- a/content/matrices/components/gaussian.ts +++ b/content/matrices/components/gaussian.ts @@ -44,15 +44,56 @@ export class Gaussian extends CustomElementView { nextStep() { console.log(`On Step ${this.stepNumber}`); const steps = [ - () => this.createBlankRightMatrix(), // 0 - () => this.drawArrow(), // 1 - () => this.copyRow(0), // 2 - () => this.copyRow(1), // 3 - () => this.moveRightMatrixToLeft(), // 4 + // Step 1: set to 1. + () => this.createBlankRightMatrix(), + () => this.multiplyRow(0, -1), + // () => this.drawArrow(0), + () => this.hideArrows(), + () => this.copyRow(0), + () => this.drawArrow(1), + // () => this.multiplyRow(1, 3), + () => this.copyRow(1), + () => this.hideArrows(), + () => this.moveRightMatrixToLeft(), + + // Step 2: multiply by 1/2 + () => this.createBlankRightMatrix(), + () => this.multiplyRow(1, 1 / 2), + () => this.drawArrow(1), + () => this.copyRow(1), + () => this.hideArrows(), + () => this.drawArrow(0), + () => this.copyRow(0), + () => this.hideArrows(), + () => this.moveRightMatrixToLeft(), + + // Step 3: Add rows 1 and 2 into row 2 + () => this.createBlankRightMatrix(), + () => this.drawArrow(0), + () => this.copyRow(0), + () => this.hideArrows(), + () => this.comboRow(0, 1, 1, 1, 1), + () => this.copyRow(1), + () => this.moveRightMatrixToLeft(), - () => this.createBlankRightMatrix(), // 5 + // Step 4: Put 1 in the diagonal. + () => this.createBlankRightMatrix(), + () => this.drawArrow(0), () => this.copyRow(0), + () => this.hideArrows(), + () => this.multiplyRow(1, -2), + () => this.drawArrow(1), () => this.copyRow(1), + () => this.hideArrows(), + () => this.moveRightMatrixToLeft(), + + // Step 5: Add Rows 1 and 2. + () => this.createBlankRightMatrix(), + () => this.comboRow(0, 1, 0, 1, 1), + () => this.copyRow(0), + () => this.drawArrow(1), + () => this.copyRow(1), + () => this.hideArrows(), () => this.moveRightMatrixToLeft() ]; steps[this.stepNumber](); @@ -169,12 +210,25 @@ export class Gaussian extends CustomElementView { const _fill = $N('rect', fillAttr, this.$rightMatrix) as SVGView; } - drawArrow() { + drawArrow(rowNumber: number) { // TODO: check fibonacci for how arrow/paths are drawn + const y = 14 + rowNumber * 50 + 37 / 2; + const $arrow1 = $N('line', {class: 'arrow', x1: 186 - 13, x2: 400 + 13, y1: y, y2: y}); + // for ordering + // FIXME: what's the right way to order these??? Ideally over the matrix but behind the rows. + this.$leftMatrix.insertBefore($arrow1); + $arrow1.hide(); + // FIXME: draw arrow head when it finishes (or draw it with?) + $arrow1.enter('draw', 200); + } + + hideArrows() { + this.$frame.$$('.arrow').forEach(a => a.remove()); + this.$frame.$$('.multiplier').forEach(a => a.remove()); } - copyRow(_rowNumber: number) { - const newRow = this.$rows[_rowNumber].copy(true, false); + copyRow(rowNumber: number) { + const newRow = this.$rows[rowNumber].copy(true, false); this.$rightMatrix.append(newRow); newRow.animate({transform: [ `translate(0px, 0px)`, @@ -182,12 +236,37 @@ export class Gaussian extends CustomElementView { ]}, 800, 500); } - multiplyRow(_rowNumber: number, _multiplier: number) { + multiplyRow(rowNumber: number, multiplier: number) { // TODO: draw arrow with a multiplier circle + this.drawArrow(rowNumber); + const cx = (186 + (400 - 186) / 2); + const cy = 14 + rowNumber * 50 + 37 / 2; + const r = 15; + const $circle = $N('circle', {class: 'multiplier', cx, cy, r, fill: 'red'}, this.$frame); + const $label = $N('text', {class: 'multiplier', x: cx, y: cy, fill: 'white', 'text-anchor': 'middle', 'dominant-baseline': 'central', text: `x${multiplier}`}, this.$frame); + + const $text = this.$rows[rowNumber].$$('text'); + const numbers = this.rows[rowNumber]; + const newNumbers = numbers.map(n => n * multiplier); + newNumbers.forEach((n, i) => { + $text[i].textStr = n; + this.rows[rowNumber][i] = n; + }); + } comboRow(_rowSrc1: number, _rowSrc2: number, _rowDest: number, _m1: number, _m2: number) { // TODO: draw multiple arrows into one + + const src1 = this.rows[_rowSrc1].map(s1 => s1 * _m1); + const src2 = this.rows[_rowSrc2].map(s2 => s2 * _m2); + const newNumbers = src1.map((s1, i) => s1 + src2[i]); + + const $text = this.$rows[_rowDest].$$('text'); + newNumbers.forEach((n, i) => { + $text[i].textStr = n; + this.rows[_rowDest][i] = n; + }); } moveRightMatrixToLeft() { diff --git a/content/matrices/styles.less b/content/matrices/styles.less index 07843d967..e1cbf2ff2 100644 --- a/content/matrices/styles.less +++ b/content/matrices/styles.less @@ -130,6 +130,11 @@ rect.row-fill { fill: @blue; } +x-gaussian line { + stroke: black; + stroke-width: 2px; +} + text.row-text { fill: '#FFFFFF'; font-size: 20; From cc847059010a7a98010a81b60d2b9398d660e05f Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Mon, 7 Dec 2020 07:32:39 -0500 Subject: [PATCH 14/20] Gaussian: beginning to codify 3x3 matrix solution steps --- content/matrices/components/gaussian.ts | 76 +++----------- content/matrices/functions.ts | 131 ++++++++++++++++++++++-- content/matrices/styles.less | 12 ++- 3 files changed, 146 insertions(+), 73 deletions(-) diff --git a/content/matrices/components/gaussian.ts b/content/matrices/components/gaussian.ts index acd24415c..da93c8754 100644 --- a/content/matrices/components/gaussian.ts +++ b/content/matrices/components/gaussian.ts @@ -26,6 +26,7 @@ export class Gaussian extends CustomElementView { private $rows!: SVGView[]; private stepNumber = 0; + private steps: (() => void)[]; ready() { console.log('This is the Gaussian'); @@ -41,62 +42,13 @@ export class Gaussian extends CustomElementView { this.displayRows(); } + setNextSteps(stepList: (() => void)[]) { + this.steps = stepList; + } + nextStep() { console.log(`On Step ${this.stepNumber}`); - const steps = [ - // Step 1: set to 1. - () => this.createBlankRightMatrix(), - () => this.multiplyRow(0, -1), - // () => this.drawArrow(0), - () => this.hideArrows(), - () => this.copyRow(0), - () => this.drawArrow(1), - // () => this.multiplyRow(1, 3), - () => this.copyRow(1), - () => this.hideArrows(), - () => this.moveRightMatrixToLeft(), - - // Step 2: multiply by 1/2 - () => this.createBlankRightMatrix(), - () => this.multiplyRow(1, 1 / 2), - () => this.drawArrow(1), - () => this.copyRow(1), - () => this.hideArrows(), - () => this.drawArrow(0), - () => this.copyRow(0), - () => this.hideArrows(), - () => this.moveRightMatrixToLeft(), - - // Step 3: Add rows 1 and 2 into row 2 - () => this.createBlankRightMatrix(), - () => this.drawArrow(0), - () => this.copyRow(0), - () => this.hideArrows(), - () => this.comboRow(0, 1, 1, 1, 1), - () => this.copyRow(1), - () => this.moveRightMatrixToLeft(), - - // Step 4: Put 1 in the diagonal. - () => this.createBlankRightMatrix(), - () => this.drawArrow(0), - () => this.copyRow(0), - () => this.hideArrows(), - () => this.multiplyRow(1, -2), - () => this.drawArrow(1), - () => this.copyRow(1), - () => this.hideArrows(), - () => this.moveRightMatrixToLeft(), - - // Step 5: Add Rows 1 and 2. - () => this.createBlankRightMatrix(), - () => this.comboRow(0, 1, 0, 1, 1), - () => this.copyRow(0), - () => this.drawArrow(1), - () => this.copyRow(1), - () => this.hideArrows(), - () => this.moveRightMatrixToLeft() - ]; - steps[this.stepNumber](); + this.steps[this.stepNumber](); this.stepNumber++; // = this.stepNumber + 1; } @@ -148,6 +100,7 @@ export class Gaussian extends CustomElementView { const allTextAttr = { class: 'row-text', + 'text-anchor': 'middle', 'dominant-baseline': 'central', fill: 'white' }; @@ -159,7 +112,8 @@ export class Gaussian extends CustomElementView { const divX = this.numVars === 2 ? 129 : 175; const dividerAttr = { - x1: divX, x2: divX, stroke: 'white', 'stroke-width': '2', 'stroke-dasharray': '2 2' + x1: divX, x2: divX, + class: 'divider' }; const _rowFill = $N('rect', Object.assign({y}, fillAttr), $row) as SVGView; @@ -233,7 +187,7 @@ export class Gaussian extends CustomElementView { newRow.animate({transform: [ `translate(0px, 0px)`, `translate(400px, 0px)` - ]}, 800, 500); + ]}, 800); } multiplyRow(rowNumber: number, multiplier: number) { @@ -242,12 +196,12 @@ export class Gaussian extends CustomElementView { const cx = (186 + (400 - 186) / 2); const cy = 14 + rowNumber * 50 + 37 / 2; const r = 15; - const $circle = $N('circle', {class: 'multiplier', cx, cy, r, fill: 'red'}, this.$frame); - const $label = $N('text', {class: 'multiplier', x: cx, y: cy, fill: 'white', 'text-anchor': 'middle', 'dominant-baseline': 'central', text: `x${multiplier}`}, this.$frame); + const _$circle = $N('circle', {class: 'multiplier', cx, cy, r, fill: 'red'}, this.$frame); + const _$label = $N('text', {class: 'multiplier', x: cx, y: cy, fill: 'white', 'text-anchor': 'middle', 'dominant-baseline': 'central', text: `x${multiplier}`}, this.$frame); const $text = this.$rows[rowNumber].$$('text'); const numbers = this.rows[rowNumber]; - const newNumbers = numbers.map(n => n * multiplier); + const newNumbers = numbers.map(n => Math.round(n * multiplier * 100) / 100); newNumbers.forEach((n, i) => { $text[i].textStr = n; this.rows[rowNumber][i] = n; @@ -258,8 +212,8 @@ export class Gaussian extends CustomElementView { comboRow(_rowSrc1: number, _rowSrc2: number, _rowDest: number, _m1: number, _m2: number) { // TODO: draw multiple arrows into one - const src1 = this.rows[_rowSrc1].map(s1 => s1 * _m1); - const src2 = this.rows[_rowSrc2].map(s2 => s2 * _m2); + const src1 = this.rows[_rowSrc1].map(s1 => Math.round(s1 * _m1 * 100) / 100); + const src2 = this.rows[_rowSrc2].map(s2 => Math.round(s2 * _m2 * 100) / 100); const newNumbers = src1.map((s1, i) => s1 + src2[i]); const $text = this.$rows[_rowDest].$$('text'); diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index a734db7df..5fc05cbe8 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -379,11 +379,64 @@ export function rotationInverse(_$step: Step) { export function gaussianSolve(_$step: Step) { const $gauss = _$step.$('x-gaussian') as Gaussian; + $gauss.setNextSteps( + [ + // Step 1: set to 1. + () => $gauss.createBlankRightMatrix(), + () => $gauss.multiplyRow(0, -1), + // () => $gauss.drawArrow(0), + () => $gauss.hideArrows(), + () => $gauss.copyRow(0), + () => $gauss.drawArrow(1), + // () => $gauss.multiplyRow(1, 3), + () => $gauss.copyRow(1), + () => $gauss.hideArrows(), + () => $gauss.moveRightMatrixToLeft(), + + // Step 2: multiply by 1/2 + () => $gauss.createBlankRightMatrix(), + () => $gauss.multiplyRow(1, 1 / 2), + () => $gauss.drawArrow(1), + () => $gauss.copyRow(1), + () => $gauss.hideArrows(), + () => $gauss.drawArrow(0), + () => $gauss.copyRow(0), + () => $gauss.hideArrows(), + () => $gauss.moveRightMatrixToLeft(), + + // Step 3: Add rows 1 and 2 into row 2 + () => $gauss.createBlankRightMatrix(), + () => $gauss.drawArrow(0), + () => $gauss.copyRow(0), + () => $gauss.hideArrows(), + () => $gauss.comboRow(0, 1, 1, 1, 1), + () => $gauss.copyRow(1), + () => $gauss.moveRightMatrixToLeft(), + + // Step 4: Put 1 in the diagonal. + () => $gauss.createBlankRightMatrix(), + () => $gauss.drawArrow(0), + () => $gauss.copyRow(0), + () => $gauss.hideArrows(), + () => $gauss.multiplyRow(1, -2), + () => $gauss.drawArrow(1), + () => $gauss.copyRow(1), + () => $gauss.hideArrows(), + () => $gauss.moveRightMatrixToLeft(), + + // Step 5: Add Rows 1 and 2. + () => $gauss.createBlankRightMatrix(), + () => $gauss.comboRow(0, 1, 0, 1, 1), + () => $gauss.copyRow(0), + () => $gauss.drawArrow(1), + () => $gauss.copyRow(1), + () => $gauss.hideArrows(), + () => $gauss.moveRightMatrixToLeft() + ] + ); + const back = $gauss.$('.btn.back'); const next = $gauss.$('.btn.next'); - // FIXME: create a "step-setter" function for Gaussians - // FIXME: possibly copy the cheesecake thing in how the next steps work - // $gauss.setSteps({}); back?.on('click', () => { $gauss.backStep(); @@ -392,11 +445,71 @@ export function gaussianSolve(_$step: Step) { next?.on('click', () => { $gauss.nextStep(); }); +} - // TODO: pseudocode - const steps = [ - 'new', - 'r1.r1', - 'r2=0.5*r2' - ]; +export function gaussianSolve2($step: Step) { + const $gauss = $step.$('x-gaussian') as Gaussian; + + $gauss.setNextSteps( + [ + // Step 1: set to 1. + () => $gauss.createBlankRightMatrix(), + () => $gauss.multiplyRow(0, 1 / 168), + // () => $gauss.drawArrow(0), + () => $gauss.hideArrows(), + () => $gauss.copyRow(0), + + () => $gauss.drawArrow(1), + () => $gauss.copyRow(1), + () => $gauss.hideArrows(), + + () => $gauss.drawArrow(2), + () => $gauss.copyRow(2), + () => $gauss.hideArrows(), + + () => $gauss.moveRightMatrixToLeft(), + + // Step 2: multiply by 1/168 + () => $gauss.createBlankRightMatrix(), + () => $gauss.multiplyRow(1, 1 / 168), + () => $gauss.drawArrow(1), + () => $gauss.copyRow(1), + () => $gauss.hideArrows(), + + () => $gauss.drawArrow(0), + () => $gauss.copyRow(0), + () => $gauss.hideArrows(), + () => $gauss.drawArrow(2), + () => $gauss.copyRow(2), + () => $gauss.hideArrows(), + + + () => $gauss.moveRightMatrixToLeft(), + + // Step 3: Add rows 1 and 2 into row 2 + () => $gauss.createBlankRightMatrix(), + () => $gauss.drawArrow(0), + () => $gauss.copyRow(0), + () => $gauss.hideArrows(), + + () => $gauss.comboRow(0, 1, 1, 1, -1), + () => $gauss.copyRow(1), + + () => $gauss.drawArrow(2), + () => $gauss.copyRow(2), + () => $gauss.hideArrows(), + + () => $gauss.moveRightMatrixToLeft() + ] + ); + const back = $gauss.$('.btn.back'); + const next = $gauss.$('.btn.next'); + + back?.on('click', () => { + $gauss.backStep(); + }); + + next?.on('click', () => { + $gauss.nextStep(); + }); } diff --git a/content/matrices/styles.less b/content/matrices/styles.less index e1cbf2ff2..0ac183dc3 100644 --- a/content/matrices/styles.less +++ b/content/matrices/styles.less @@ -130,12 +130,18 @@ rect.row-fill { fill: @blue; } -x-gaussian line { +x-gaussian .arrow { stroke: black; stroke-width: 2px; } +x-gaussian .divider { + stroke: white; + stroke-width: 2; + stroke-dasharray: 2 2; +} + text.row-text { - fill: '#FFFFFF'; - font-size: 20; + fill: #FFFFFF; + font-size: 20px; } \ No newline at end of file From b82e878bbf1a2dc83a94bd7d3feff180207679d1 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Mon, 7 Dec 2020 07:34:21 -0500 Subject: [PATCH 15/20] Remove extra section for singular matrices --- content/matrices/content.md | 35 ----------------------------------- 1 file changed, 35 deletions(-) diff --git a/content/matrices/content.md b/content/matrices/content.md index ba16ec63a..9ab609462 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1582,41 +1582,6 @@ Here is a collection of points that have been multiplied by a **singular matrix* 📈Interactive where student can adjust a matrix that multiplies by a singular matrix, but the resulting transform will always be a line. ---- -> id: singular-lines - -### **Non-intersecting lines (singular matrices)** - -We have seen how matrices with determinant = zero do not have inverses and are called **singular matrices**. This means that when a 2x2 matrix representing two lines is singular, it means the lines [[do not have a single solution|do not intersect|are parallel]] (tricky wording). - -**Example:** these two lines are actually the same. When we rewrite their line equations in matrix multiplication form, the matrix has a determinant of zero. There is no inverse, and thus [[no unique solution|no solution]] - - figure: img(src="images/proto-4/Untitled%206.png") - -Two lines y = -x + 3 and -4y = 4x - 12, are actually the same line - - figure: img(src="images/proto-4/Untitled%207.png") - -two equations - - figure: img(src="images/proto-4/Untitled%208.png") - -equations rewritten as a matrix - -**Example:** these two lines are parallel. When we rewrite their line equations in matrix multiplication form, it has a determinant of zero. There is no inverse, and thus [[no unique solution]] - - figure: img(src="images/proto-4/Untitled%209.png") - -Two parallel lines y=3x+2, y=3x-4, have no intersection - - figure: img(src="images/proto-4/Untitled%2010.png") - -two line equations - - figure: img(src="images/proto-4/Untitled%2011.png") - -equations rewritten as a matrix - --- > id: least-squares From 79b0623510078d843f1915c25adac1fd1652d33d Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Mon, 7 Dec 2020 14:24:22 -0500 Subject: [PATCH 16/20] Formula for inverse of 3x3 matrix --- content/matrices/content.md | 21 ++++++++++++++++----- 1 file changed, 16 insertions(+), 5 deletions(-) diff --git a/content/matrices/content.md b/content/matrices/content.md index 9ab609462..ab2a2a202 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1667,16 +1667,27 @@ this is truly magic. To learn more about how this works, check the course on sta ::: --- -> id: inverse-conclusion +> id: three-by-three -### Conclusion/Cliff-Hanger +#### Larger Matrices + +::: column(width=300) Here is the formula for calculating a 3x3 matrix. There is a way to calculate inverses of larger matrices, which we will observe next chapter. -{.fixme} Formula for inverses of 3x3 matrices (and maybe the pattern for larger matrices) +::: column.grow + + +`§[[a b c] [d e f] [g h i]]^(-1)` = `1/det(A)` • `§[[(ei-fh) (ch-bi) (bf-ce)] [(fg-di) (ai-cg) (cd-af)] [(dh-eg) (bg-ah) (ae-bd)]]` + +::: + + +--- +> id: inverse-conclusion + +### Conclusion/Cliff-Hanger -- Maybe a large formula? -- Maybe a list of steps that show what to do. "Throughout history mathematicians have done many ways to solve inverses... for example (statistics and orbit of planets). But our current methods are not enough to solve these, we need something better!" From 930cda11e2eabec5f4520b52f1db50586e6e1d8b Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Sat, 23 Jan 2021 13:53:55 -0500 Subject: [PATCH 17/20] guass map of netherlands --- content/matrices/content.md | 12 + content/matrices/functions.ts | 24 ++ content/matrices/svg/map-with-dots.svg | 407 +++++++++++++++++++++++++ 3 files changed, 443 insertions(+) create mode 100644 content/matrices/svg/map-with-dots.svg diff --git a/content/matrices/content.md b/content/matrices/content.md index ab2a2a202..3fc99e955 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1700,6 +1700,18 @@ Here is the formula for calculating a 3x3 matrix. There is a way to calculate in > section: systems > sectionStatus: dev +> id: cartography + +Here's a map + + figure: include svg/map-with-dots.svg + + +A map of Netherlands that Gauss helped fix. + +--- + + {.todo} COMING SOON > id: gaussian-solve diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index 5fc05cbe8..5d531e95a 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -15,6 +15,30 @@ import './components/gaussian'; import {Gaussian} from './components/gaussian'; +export function map($step: Step) { + console.log('map'); + const $map = $step.$('#map-with-dots'); + const $netherlands = $step.$('#netherlands'); + // const $linesKeep = $step.$('#lines-keep'); + const $linesHide = $step.$('#lines-hide'); + const $mapHide = $step.$('#points-map-hide'); + // const $mapShow = $step.$('#points-map-show'); + + console.log($map); + setTimeout(() => { + $netherlands?.exit('fade'); + }, 2000); + + setTimeout(() => { + $linesHide?.exit('fade'); + $mapHide?.exit('fade'); + }, 3000); + + setTimeout(() => { + $map?.css('transform', 'scale(2) translate(-50%,0)'); + }, 6000); +} + export function rocket($step: Step) { const $geo = $step.$('x-geopad') as Geopad; $step.model.th = 0; diff --git a/content/matrices/svg/map-with-dots.svg b/content/matrices/svg/map-with-dots.svg new file mode 100644 index 000000000..70aa2f899 --- /dev/null +++ b/content/matrices/svg/map-with-dots.svg @@ -0,0 +1,407 @@ + + + map-with-dots + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file From 1d2016b9c6bca51eb3979a2ea3fce09e4fe1cf01 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Wed, 27 Jan 2021 09:56:42 -0500 Subject: [PATCH 18/20] prepping for code --- content/matrices/content.md | 16 +++++++++ content/matrices/functions.ts | 63 ++++++++++++++++++++++++++++++----- 2 files changed, 71 insertions(+), 8 deletions(-) diff --git a/content/matrices/content.md b/content/matrices/content.md index 3fc99e955..d7c5eee8a 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1700,6 +1700,22 @@ Here is the formula for calculating a 3x3 matrix. There is a way to calculate in > section: systems > sectionStatus: dev +> id: babylon + +Here is a Gaussian component + +::: x-gaussian(dimension="2x2") + +{div(slot="matrix")} `§[[1 1] [(2/3) (-1/2)]]` • `§[[x] [y]]` = `§[[1800] [500]]` + +* Here is step one +* Here is step two +* Here is step three + +::: + +--- + > id: cartography Here's a map diff --git a/content/matrices/functions.ts b/content/matrices/functions.ts index 5d531e95a..cb664d389 100644 --- a/content/matrices/functions.ts +++ b/content/matrices/functions.ts @@ -400,12 +400,56 @@ export function rotationInverse(_$step: Step) { // TODO: put an x-matrix-solver here } +class LinearCombinationStep { + + constructor( + public multiply: number[], + public into: number, + public description: string) { + // + } +} + +export function babylon($step: Step) { + const $gaussian = $step.$('x-gaussian') as Gaussian; + + const linearCombinationSteps = [ + new LinearCombinationStep([-2 / 3, 1], 2, 'do a thing'), + new LinearCombinationStep([0, -6 / 7], 2, 'simplify to solve R2'), + new LinearCombinationStep([1, -1], 1, 'back substitute') + ]; + + // -- how to figure out this part? Step-by-step... + const _functionSteps = linearCombinationSteps.map(lc => { + + const multiplySteps = lc.multiply.map((multiplier, i) => { + return [ + () => $gaussian.createBlankRightMatrix(), + () => $gaussian.multiplyRow(multiplier, i - 1), + () => $gaussian.hideArrows(), + () => $gaussian.copyRow(0), // another loop + () => $gaussian.drawArrow(i - 1), + () => $gaussian.copyRow(1), // should be i - 1 + () => $gaussian.hideArrows(), + () => $gaussian.moveRightMatrixToLeft() + ]; + }); + + return multiplySteps; + }); + /// MAP AND APPEND OR SOMETHING + // $gaussian.setNextSteps(functionSteps); // why??? type not work??? +} + +// ========================== +// === GAUSSIAN SOLVER ==== +// ========================== export function gaussianSolve(_$step: Step) { const $gauss = _$step.$('x-gaussian') as Gaussian; $gauss.setNextSteps( [ - // Step 1: set to 1. + // -- Step 1: set to 1. () => $gauss.createBlankRightMatrix(), () => $gauss.multiplyRow(0, -1), // () => $gauss.drawArrow(0), @@ -417,7 +461,7 @@ export function gaussianSolve(_$step: Step) { () => $gauss.hideArrows(), () => $gauss.moveRightMatrixToLeft(), - // Step 2: multiply by 1/2 + // -- Step 2: multiply by 1/2 () => $gauss.createBlankRightMatrix(), () => $gauss.multiplyRow(1, 1 / 2), () => $gauss.drawArrow(1), @@ -428,7 +472,7 @@ export function gaussianSolve(_$step: Step) { () => $gauss.hideArrows(), () => $gauss.moveRightMatrixToLeft(), - // Step 3: Add rows 1 and 2 into row 2 + // -- Step 3: Add rows 1 and 2 into row 2 () => $gauss.createBlankRightMatrix(), () => $gauss.drawArrow(0), () => $gauss.copyRow(0), @@ -437,7 +481,7 @@ export function gaussianSolve(_$step: Step) { () => $gauss.copyRow(1), () => $gauss.moveRightMatrixToLeft(), - // Step 4: Put 1 in the diagonal. + // -- Step 4: Put 1 in the diagonal. () => $gauss.createBlankRightMatrix(), () => $gauss.drawArrow(0), () => $gauss.copyRow(0), @@ -448,7 +492,7 @@ export function gaussianSolve(_$step: Step) { () => $gauss.hideArrows(), () => $gauss.moveRightMatrixToLeft(), - // Step 5: Add Rows 1 and 2. + // -- Step 5: Add Rows 1 and 2. () => $gauss.createBlankRightMatrix(), () => $gauss.comboRow(0, 1, 0, 1, 1), () => $gauss.copyRow(0), @@ -471,12 +515,15 @@ export function gaussianSolve(_$step: Step) { }); } +// ========================== +// === GAUSSIAN SOLVER ==== +// ========================== export function gaussianSolve2($step: Step) { const $gauss = $step.$('x-gaussian') as Gaussian; $gauss.setNextSteps( [ - // Step 1: set to 1. + // -- Step 1: set to 1. () => $gauss.createBlankRightMatrix(), () => $gauss.multiplyRow(0, 1 / 168), // () => $gauss.drawArrow(0), @@ -493,7 +540,7 @@ export function gaussianSolve2($step: Step) { () => $gauss.moveRightMatrixToLeft(), - // Step 2: multiply by 1/168 + // -- Step 2: multiply by 1/168 () => $gauss.createBlankRightMatrix(), () => $gauss.multiplyRow(1, 1 / 168), () => $gauss.drawArrow(1), @@ -510,7 +557,7 @@ export function gaussianSolve2($step: Step) { () => $gauss.moveRightMatrixToLeft(), - // Step 3: Add rows 1 and 2 into row 2 + // -- Step 3: Add rows 1 and 2 into row 2 () => $gauss.createBlankRightMatrix(), () => $gauss.drawArrow(0), () => $gauss.copyRow(0), From 803d39518bf5ef6357c42085531910da965b1b0c Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Wed, 27 Jan 2021 11:45:11 -0500 Subject: [PATCH 19/20] First pass at copying g-doc into markdown --- content/matrices/content.md | 167 +++++++++++++++++++++++++++++++++++- 1 file changed, 164 insertions(+), 3 deletions(-) diff --git a/content/matrices/content.md b/content/matrices/content.md index d7c5eee8a..a43e9878a 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1700,7 +1700,68 @@ Here is the formula for calculating a 3x3 matrix. There is a way to calculate in > section: systems > sectionStatus: dev -> id: babylon +> id: gauss-intro-piazzi + +### Intro + +The year is 1801. Giuseppe Piazzi is an Italian astronomer who has catalogued the precise position of over 7,000 stars. He published the Catalog of Stars, observed from the Palermo observatory. + +On January 2nd, 1801, Piazzi observed a celestial body[1] whose location did not match his previous observation. Had Piazzi committed an error in measurement, or had this body moved through the night sky? He repeated the observation over the next two days and surely enough it had moved across the sky. The next day Piazzi could not make another measurement because this unknown entity had moved behind the sun. + +The mathematics of the time were not sufficient to calculate the orbit of this body using only three points. Fortunately, Friedrich Gauss[2] learned of this predicament and took on the challenge. He invented a new method to predict the orbit of this celestial body and predicted its location in December of 1801. Surely enough, it was found exactly where he predicted it would be! The body was found to be a dwarf planet with an orbit between Mars and Jupiter. It was named Ceres, after a Roman goddess of agriculture. + +--- + +> id: ceres-orbit + +### Orbit of Ceres + +To calculate Ceres’ orbit around the sun, Gauss needed to do some complex math with triangles. The three Es here represent the locations of the Earth as it moves around the sun, the lines labeled L represent the line of sight that Gauss measured, and the positions P1, P2, P3 are the calculated positions of Ceres relative to the sun. + +{.todo} IMAGE_NEEDED of diagrams + +The goal was to replace the three positions of Ceres with parameters that represented the orbit around the sun.[3] When searching for these positions of Ceres, Gauss encountered a mathematical problem with a very long history, that dates back to almost 4,000 years ago! + + +--- + +> id: babylon-intro + +### Ancient Babylonians: 2x2 + +The problem of how to calculate the orbit of Ceres can ultimately be considered a system of equations. What if you have multiple unknown variables, and multiple equations that represent the relationships between those unknowns? This was not a new problem - in fact, it dates back to at least the ancient Babylonians in 1800 BCE. + +The Babylonians[4][5] used Cuneiform tablets to teach algebra. Their culture and society was very dependent on agriculture. Here is a problem that was found inscribed on a tablet. It was translated by historians/linguists. + +{.todo} IMAGE_NEEDED of Cuneiform + +--- + +> id: babylon-setup + +::: .box.f-green + +I have two fields that total 1800 sar (square meters). Yields for each field are ⅔ kur of grain per sar and ½ kur of grain per sar. The first field gave 500 more kur than the second. What are the areas of each field?[4][5] + +::: + +The units are unfamiliar, but we just need to know what kind of unit they represent (area, volume/mass) and we can repeat the techniques with whatever units we choose. + +Now let’s follow some steps to solve this system of equations: + + +--- + +> id: babylon-problem + +### Babylon Problem + +::: .box.f-red + +{.todo} PROBLEM_SOLVE details needed of problem + +::: + Here is a Gaussian component @@ -1714,16 +1775,116 @@ Here is a Gaussian component ::: +Great! The problem we just solved was likely solved by maths students in an ancient civilization! Let’s see what else we can do with this. + +--- + +> id: chinese-intro + +### Ancient Chinese: 3x3 + +Interestingly, the ancient Chinese also used a similar method to calculate agricultural yields (a lot more people were farmers back then!). This example was found in Jiuzhang Suanshu, or Nine Chapters of the Mathematical Art. + +Instead of measuring grain, this problem measures rice, from rice paddies. The area of the volume is measured in dou, and the yield is a sheaves, which is a bundle of rice stalks. + +{.todo} IMAGE_NEEDED of rice paddies + + +--- + +> id: chinese-problem + +Here is the problem: + +::: .box.f-green + +There are three grades of rice paddies, top-grade, medium-grade, and low-grade. The combined yield of the first paddy is 39 dou of grain for 3 sheaves from top-grade paddies, 2 from medium-grade, and 1 from low-grade; and similarly for two more collections of sheaves. The other two rice paddies have the yields shown in the table. + +**What is the expected yield from one sheave from a top-grade field, a medium-grade field, and a low-grade field?** + +::: + + +Here is a Gaussian component: + +::: .box.f-red + +{.todo} PROBLEM_SOLVE details needed of problem + +::: + +What we just did with the help of computers, Ancient Chinese Mathematicians actually used a table with counting rods (similar to an abacus). Before they had computers, they found a way to do this! + --- -> id: cartography +> id: rules-gaussian + +### Rules of Gaussian Elimination + +Great job! We just worked through two problems that were likely solved by maths students in ancient civilizations! These kinds of problems have troubled mathematicians for ages. Despite being used long ago, the method we used in those two problems is what we now call [Gaussian Elimination](gloss:gaussian-elimination). + +Gaussian Elimination is used to solve a system of equations. It is a procedure in which a system of equations can be written as a matrix problem, and then simplified to find solutions for variables. + +::: .box.f-yellow + +**Process** + +- Write the system of equations as an augmented matrix form. +- Follow the steps until the identity matrix is reached + - Get zeros in the bottom-left diagonal of the matrix. (This is called upper Triangle matrix) + - Replace each row with a linear combination of that row, and the row above it. Choose a linear combination so that the leading coefficient is zero. ((NOTE: this does NOT add new information or change the meaning of the equations. It is like performing the same operation on multiple sides of an equation)). + - Divide to make the leading coefficient 1. + - Perform “back-substitution” to get zeros in the top-right diagonal of the matrix. This will leave you with the identity matrix. + +::: + + +--- + +> id: solving-ceres + +### Ceres + +Solving Ceres + + +--- + +> id: cartography-intro + +### Cartography + +Gauss also used this approach to solve other problems people were struggling with. Here's another example. At the turn of the 19th century there was a large trend of national governments wanting accurate geographical maps of their country, for purposes of military safety, navigation, and urban planning. The land was mapped by picking landmarks throughout the country. At each location, a “surveyor” would establish a line of sight from their location to two other landmarks. They then measured angles between the lines of sight. + +{.todo} IMAGE_NEEDED of landmarks & measuring + +These landmarks were spread across very large distances, so measuring the angle off by even a degree could send the point off by many miles. Thus, it was essential to get the measurements as close as possible. Also, many points were inaccessible due to being on mountains. + Here's a map figure: include svg/map-with-dots.svg +Here you can see all the measurements made by surveyors working for the Dutch government in 1826. Here is a map of Holland, with many landmarks picked out. The yellow lines are the lines of sight between landmarks. + +The problem we will do below is only a slight approximation to what Gauss actually did, which was to use a method called Least Squares, which is out of scope for this chapter. Least Squares is not for solving but for minimizing total error over many variables. Gauss used this method of least squares to minimize the error over all of the measurements. Imagine how complex it would be to minimize the error over a map with dozens of locations and hundreds of angles. Gauss’s methods of elimination were essential to devising accurate maps. + + + +--- + +> id: cartography-problem + +### Cartography Problem + +Let's simplify this problem by picking just 6 landmarks and some angles between them. + +::: .box.f-green + +{.todo} PROBLEM_SOLVE details needed of problem + +::: -A map of Netherlands that Gauss helped fix. --- From 2a8c0a6382964bc8bb5b4232c9ade0d55894bf73 Mon Sep 17 00:00:00 2001 From: Kevin DeLand Date: Mon, 8 Feb 2021 15:02:32 -0500 Subject: [PATCH 20/20] todo --- content/matrices/content.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/content/matrices/content.md b/content/matrices/content.md index a43e9878a..29ed93018 100644 --- a/content/matrices/content.md +++ b/content/matrices/content.md @@ -1708,6 +1708,8 @@ The year is 1801. Giuseppe Piazzi is an Italian astronomer who has catalogued th On January 2nd, 1801, Piazzi observed a celestial body[1] whose location did not match his previous observation. Had Piazzi committed an error in measurement, or had this body moved through the night sky? He repeated the observation over the next two days and surely enough it had moved across the sky. The next day Piazzi could not make another measurement because this unknown entity had moved behind the sun. +{.todo} IMAGE_NEEDED of Piazzi? Telescope? Planets? + The mathematics of the time were not sufficient to calculate the orbit of this body using only three points. Fortunately, Friedrich Gauss[2] learned of this predicament and took on the challenge. He invented a new method to predict the orbit of this celestial body and predicted its location in December of 1801. Surely enough, it was found exactly where he predicted it would be! The body was found to be a dwarf planet with an orbit between Mars and Jupiter. It was named Ceres, after a Roman goddess of agriculture. ---