Matrix from visual representation of transformation

Let's practice encoding linear transformations as matrices, as described in the previous article. For instance, suppose we want to find a matrix which corresponds with a 90${}^{\circ }$‍ rotation.

The first column of the matrix tells us where the vector $\left[\begin{array}{c}1\\ 0\end{array}\right]$‍ goes, and—looking at the animation—we see that this vector lands on $\left[\begin{array}{c}0\\ 1\end{array}\right]$‍. Based on this knowledge, we start filling in our matrix like this:

For the second column, we ask where the vector $\left[\begin{array}{c}0\\ 1\end{array}\right]$‍ lands. Rotating this upward facing vector 90${}^{\circ }$‍ yields a leftward facing arrow—i.e., the vector $\left[\begin{array}{c}-1\\ 0\end{array}\right]$‍—so we can finish writing our matrix as $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$‍.

Now you try!

Problem 1

What matrix corresponds with the following transformation?

Problem 2

What matrix corresponds with the following transformation?

Problem 3

What matrix corresponds with the following transformation?

Problem 4

What matrix corresponds with the following transformation?

Problem 5

What matrix corresponds with the following transformation?

Problem 6

What matrix corresponds with the following transformation?