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 90degree rotation.
The first column of the matrix tells us where the vector [10]\greenD{\left[ \begin{array}{c} 1 \\ 0 \end{array} \right]} goes, and—looking at the animation—we see that this vector lands on [01]\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]. Based on this knowledge, we start filling in our matrix like this:
[0?1?] \left[ \begin{array}{cc} 0 & ? \\ 1 & ? \end{array} \right]
For the second column, we ask where the vector [01]\redD{\left[ \begin{array}{c} 0 \\ 1 \end{array} \right]} lands. Rotating this upward facing vector 90degree yields a leftward facing arrow—i.e., the vector [10]\left[ \begin{array}{c} -1 \\ 0 \end{array} \right]—so we can finish writing our matrix as [0110] \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?
Choose 1 answer:
Choose 1 answer:

Problem 2
What matrix corresponds with the following transformation?
Choose 1 answer:
Choose 1 answer:

Problem 3
What matrix corresponds with the following transformation?
Choose 1 answer:
Choose 1 answer:

Problem 4
What matrix corresponds with the following transformation?
Choose 1 answer:
Choose 1 answer:

Problem 5
What matrix corresponds with the following transformation?
Choose 1 answer:
Choose 1 answer:

Problem 6
What matrix corresponds with the following transformation?
Choose 1 answer:
Choose 1 answer: