uplakanimkk

2022-07-14

Suppose $T$ is a transformation from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$. Find the matrix $A$ that induces $T$ if $T$ is the (counter-clockwise) rotation by $\frac{3}{4}\pi$.
how to begin to find a matrix that is 2x2 for this question.

Dayana Zuniga

The rotation matrix is $\left[\begin{array}{cc}\mathrm{cos}\left(\theta \right)& -\mathrm{sin}\left(\theta \right)\\ \mathrm{sin}\left(\theta \right)& \mathrm{cos}\left(\theta \right)\end{array}\right]$But blindly applying this formula doesn't teach you the most important part of linear transformations. (ANY linear transformation matrix is defined by where it takes the unit vectors.)
This linear transformation T that is a counterclockwise rotation takes $\left[\begin{array}{c}1\\ 0\end{array}\right]$ to $\left[\begin{array}{c}\mathrm{cos}\left(\theta \right)\\ \mathrm{sin}\left(\theta \right)\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$ to $\left[\begin{array}{c}-\mathrm{sin}\left(\theta \right)\\ \mathrm{cos}\left(\theta \right)\end{array}\right]$. You can confirm this on your own using geometry.
And this works for ANY linear transformation.

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