Matrix from visual representation of transformation

Learn how to determine the transformation matrix that has a given effect that is described visually.

Warmup example

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 $\greenD{\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]}$ 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:
$\left[ \begin{array}{cc} 0 & ? \\ 1 & ? \end{array} \right]$
For the second column, we ask where the vector $\redD{\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]}$ 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 $\left[ \begin{array}{cc} 0 & \redD{-1} \\ 1 & \redD{0} \end{array} \right]$.
Now you try!

Practice problems

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?