determine the transformation matrix rotation

Question

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 $\tfrac{3}{4}\pi$.

I am not sure how to begin to find a matrix that is $2\times 2$ for this question.

Answer 1

The rotation matrix is $\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\\sin(\theta) &\cos(\theta) \end{bmatrix}$.

---

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 $\begin{bmatrix} 1\\0 \end{bmatrix}$ to $\begin{bmatrix} \cos(\theta)\\\sin(\theta) \end{bmatrix}$ and $\begin{bmatrix} 0\\1 \end{bmatrix}$ to $\begin{bmatrix} -\sin(\theta)\\\cos(\theta) \end{bmatrix}$. You can confirm this on your own using geometry.

And this works for ANY linear transformation.