2b:null 2c:["$","title",null,{"children":"Affine Transformations | Lao | Observable"}] 31:Tf56,md` ### More Information Linear algebra is the study of vector spaces and linear transformation between vectors. For something to be a vector space, there must be a vector addition and a scalar multiplication that has these properties: Let vectors ${tex`\vec{u}, \vec{v}, \vec{w}, \vec{0}`} exist in vector space ${tex`V`} over field ${tex`\mathbb{F}`} and ${tex`1, a, b`} exist in field ${tex`\mathbb{F}`} | Vector Addition | Scalar Multiplication | | ----------------| ----------------------| | *associative*
${tex`\vec{u}+(\vec{v}+\vec{w}) = (\vec{u}+\vec{v})+\vec{w}`} | *associative*
${tex`a \cdot (b \cdot \vec{v}) = (a \cdot b) \cdot \vec{v}`} | | *commutative*
${tex`\vec{u}+\vec{v} = \vec{v}+\vec{u}`} | ${tex``} | | *identity element*
${tex`\vec{0} + \vec{v} = \vec{v}`} | *identity element*
${tex`1 \cdot \vec{v} = \vec{v}`} | | *inverse*
${tex`\vec{v} + (-\vec{v}) = \vec{0}`} | ${tex``} | | ${tex``} | *vector distribution*
${tex`a \cdot (\vec{u}+\vec{v}) = a \cdot \vec{u} + a \cdot \vec{v}`} | | ${tex``} | *field distribution*
${tex`(a+b) \cdot \vec{v} = a \cdot \vec{v} + b \cdot \vec{v}`} | In addition to vector spaces, there are also linear transformations between vector spaces that preserve the linear structure of equally spaced and parrellel lines of a vector space. Let ${tex`T`} be a linear transformation ${tex`T:V\to W`} between vector space ${tex`V`} to vector space ${tex`W`}, with vectors ${tex`\vec{u}, \vec{v}`} in ${tex`V`} and ${tex`a`} in field ${tex`\mathbb{F}`}. T is linear if and only if: | Invarience of Vector Addition | Invarience of Scalar Multiplication | | ----------------| ----------------------| | ${tex`T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v})`} | ${tex`T(a \cdot \vec{v}) = a \cdot T(\vec{v})`} | If a transformation is linear, it will always map the zero vector from the domain to the zero vector in the codomain. This means linear transformations can visually scale, shear, reflect or rotate a vector space about the origin, but they cannot move the space by translation, because this would move the origin and the zero vectors would no longer be aligned. There are two ways around this. First, the affine transformation ${tex`f(\vec{x})`} can be represented as a linear transformation ${tex`T`} of scaling, rotating and sheering plus a vector for translation ${tex`\vec{b}`} : Let ${tex`T:V \to W`} be a linear transformation between vector spaces ${tex`V, W`} over field ${tex`\mathbb{F}`} with vector ${tex`\vec{x}`} in ${tex`V`} and vector ${tex`\vec{b}`} in ${tex`W`}. An affine transformation is: ${tex`f(\vec{x}) = T\vec{x} + \vec{b}`}
A second way is to add an additional dimension and treat translation as a shear 1 unit above the origin: ${tex`f(\vec{x}) \oplus [1] = \left[\begin{array}{c|c} T & \begin{matrix} | \\ \vec{b} \\ | \end{matrix} \\ \hline 0 \space \space ... \space \space 0 & 1 \end{array}\right] \left[\begin{array}{c} | \\ \vec{x} \\ | \\ \hline 1 \end{array}\right] `}
This augmented matrix is called an [affine transformation matrix][1] and is a linear in the artificial +1 dimensional space. In special case mathimatically, but the typical use case for many geometric applications like computer graphics, if a linear translation maps the oringial vector space to itself ${tex`T:V \to V`} and is an isomorphism, then the vector space ${tex`V`} equiped with the bilinear opperation ${tex`T`} is [an algebra over a field][2] (not to be confused with algebra generally). Another way of saying this is, vector spaces are defined with vector addition and scalar multiplication ${tex`+ \space \cdot`} while an algebra is also equiped with third operation ${tex`+ \space \cdot \space \circ`} matrix composition. [1]:https://en.wikipedia.org/wiki/Transformation_matrix#Affine_transformations [2]:https://en.wikipedia.org/wiki/Vector_space#Algebras_over_fields `32:T275d,M = ({ add: function add(){ let arg = arguments let size = [] for(let h=0; h0){ nicest = score.indexOf(d3.max(score)) this.rowSwap(A,pivots+1, nicest+1) if((pivots+1) != (nicest+1)){det *= -1} // invert determinant because of row opperation det *= A[pivots][j] // scale determinant by the number a row is multiplied by this.rowScale(A, 1 / A[pivots][j], pivots+1) // normalizes pivot to 1 for(let i=pivots+1; i0){ nicest = score.indexOf(d3.max(score)) // index of first value that is the nicest of column this.rowSwap(I,pivots+1, nicest+1) // move that row to next pivot position this.rowSwap(A,pivots+1, nicest+1) this.rowScale(I, 1 / A[pivots][j], pivots+1) // normalizes pivot to 1 this.rowScale(A, 1 / A[pivots][j], pivots+1) for(let i=0; i1){ for(let i=1; i 0 ){rank+=1} } return rank } , rowAdd: function(A, c, y, x){ // adds c * row y to row x x -=1; y -=1; for(let i=0; i< A[0].length; i++){ A[x][i] += c*A[y][i] if(A[x][i] < .0000001 && A[x][i] > -.0000001){A[x][i]=0} } return A } , rowScale: function(A,c,x){ x -=1; for(let i=0; i< A[0].length; i++){ A[x][i] *= c } return A } , rowSwap: function(A,x,y){ // swaps row x with row y x -=1; y -=1; [A[x], A[y]] = [A[y], A[x]]; return A; } , rowVector: function(A,x){ let row = [] for(let j=0; j0){ nicest = score.indexOf(d3.max(score)) // index of first value that is the nicest of column this.rowSwap(A,pivots+1, nicest+1) // move that row to next pivot position this.rowScale(A, 1 / A[pivots][j], pivots+1) // normalizes pivot to 1 for(let i=pivots+1; i\n

Start with positions

x =

y =

and translate

right

\n\n`;\n form.oninput = () => form.value = [\n form.x.valueAsNumber,\n form.y.valueAsNumber,\n form.right.valueAsNumber,\n form.up.valueAsNumber\n ];\n form.oninput();\n return form;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":212,"value":"{\n let x = input[0]\n let y = input[1]\n let xTimes = input[2]\n let yTimes = input[3]\n \n let v = [[x], [y], [1]]\n let R = [[1, 0, 1], [0, 1, 0], [0, 0, 1]]\n let U = [[1, 0, 0], [0, 1, 1], [0, 0, 1]]\n \n let final = M.mult (M.power(R,xTimes) , M.power(U,yTimes) , [[x],[y],[1]] )\n \n return md`\nMove position (${x}, ${y}) by (${xTimes}, ${yTimes}) to (${final[0][0]}, ${final[1][0]})\n\n${tex` \\space\\space\\space\\space ${M.tex(R)}^{${xTimes}} ${M.tex(U)}^{${yTimes}} ${M.tex([[x],[y],[1]])}`}\n
\n${tex`= ${M.tex(M.power(R,xTimes))} ${M.tex(M.power(U,yTimes))} ${M.tex([[x],[y],[1]])}`}\n\n
\n${tex`= ${M.tex( M.mult(M.power(R,xTimes), M.power(U,yTimes)))} ${M.tex([[x],[y],[1]])} = ${M.tex( M.mult (M.power(R,xTimes) , M.power(U,yTimes) , [[x],[y],[1]] ))}`}\n`\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":10,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":144,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W33","stdlib":"1","titling":"auto"}} 30:"$2f:body"