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### Course: Algebra (all content) > Unit 20

Lesson 9: Matrices as transformations- Transforming vectors using matrices
- Use matrices to transform 3D and 4D vectors
- Transforming polygons using matrices
- Transform polygons using matrices
- Matrices as transformations
- Matrix from visual representation of transformation
- Visual representation of transformation from matrix
- Understand matrices as transformations of the plane

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# 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 ${\left[\begin{array}{c}1\\ 0\end{array}\right]}$ 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!

## 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?

**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?

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