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correct a typo in line 398 (#432)
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the fourth vector should be [−2/3 -1/3 ]^T
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li1900 authored Aug 13, 2024
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Expand Up @@ -395,7 +395,7 @@ As a last step, apply the inverse change of basis matrix, multiplied on the left

Since we could do this to any vector written in her language, first applying the change of basis matrix, then the transformation then the inverse change of basis, that composition of three matrices gives us the transformation matrix in Jennifer's language. It takes in a vector in her language, and spits out the transformed version of that vector in her language.

For this example, where Jennifer's basis vectors look like $\left[\begin{array}{c} 2 \\ 1 \end{array}\right]$ and $\left[\begin{array}{c} -1 \\ 1 \end{array}\right]$ to us, and we're translating a $90$ degree rotation, the product of these three matrices, if you work through it, has columns $\left[\begin{array}{c} 1/3 \\ 5/3 \end{array}\right]$ and $\left[\begin{array}{c} -2/3 \\ 1/3 \end{array}\right]$. So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the $90$ degree rotated version of her vector, expressed in her coordinate system.
For this example, where Jennifer's basis vectors look like $\left[\begin{array}{c} 2 \\ 1 \end{array}\right]$ and $\left[\begin{array}{c} -1 \\ 1 \end{array}\right]$ to us, and we're translating a $90$ degree rotation, the product of these three matrices, if you work through it, has columns $\left[\begin{array}{c} 1/3 \\ 5/3 \end{array}\right]$ and $\left[\begin{array}{c} -2/3 \\ -1/3 \end{array}\right]$. So if Jennifer multiplies that matrix by the coordinates of a vector in her system, it will return the $90$ degree rotated version of her vector, expressed in her coordinate system.

<Figure
image="./figures/translating-transformations/TranslateMatrix.light.svg"
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