# TBIL Activities for Understanding Linear Algebra

## Section2.6The geometry of matrix transformations

### Activity2.6.0.1.

For this activity, use the applet in Understanding Linear Algebra, Activity 2.6.2 5 .
For the following matrices $$A$$ given below, use the diagram to study the effect of the corresponding matrix transformation $$T(\xvec) = A\xvec\text{.}$$ For each transformation, describe the geometric effect of the transformation on the plane.

#### (a)

The matrix $$A=\left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.}$$

#### (b)

The matrix $$A=\left[\begin{array}{rr} 2 \amp 0 \\ 0 \amp 2 \\ \end{array}\right]\text{.}$$

#### (c)

The matrix $$A=\left[\begin{array}{rr} 0 \amp -1 \\ 1 \amp 0 \\ \end{array}\right]\text{.}$$

#### (d)

The matrix $$A=\left[\begin{array}{rr} 1 \amp 1 \\ 0 \amp 1 \\ \end{array}\right]\text{.}$$

#### (e)

The matrix $$A=\left[\begin{array}{rr} -1 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.}$$

#### (f)

The matrix $$A=\left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 0 \\ \end{array}\right]\text{.}$$

#### (g)

The matrix $$A=\left[\begin{array}{rr} 1 \amp 0 \\ 0 \amp 1 \\ \end{array}\right]\text{.}$$

#### (h)

The matrix $$A=\left[\begin{array}{rr} 1 \amp -1 \\ -2 \amp 2 \\ \end{array}\right]\text{.}$$

### Activity2.6.0.2.

In this activity, we seek to describe some matrix transformations by finding the matrix that gives the desired transformation. All of the transformations that we study here have the form $$T:\real^2\to\real^2\text{.}$$

#### (a)

Find the matrix of the transformation that has no effect on vectors; that is, $$T(\xvec) = \xvec\text{.}$$ We call this matrix the identity and denote it by $$I\text{.}$$

#### (b)

Find the matrix of the transformation that reflects vectors in $$\real^2$$ over the line $$y=x\text{.}$$

#### (c)

What is the result of composing the reflection you found in the previous part with itself; that is, what is the effect of reflecting in the line $$y=x$$ and then reflecting in this line again? Provide a geometric explanation for your result as well as an algebraic one obtained by multiplying matrices.

### Activity2.6.0.3.

As above, we can describe various matrix transformations by finding the matrix that gives the desired transformation. All of the transformations that we study here have the form $$T:\real^2\to\real^2\text{.}$$

#### (a)

Find the matrix that rotates vectors counterclockwise in the plane by $$90^\circ\text{.}$$

#### (b)

Compare the result of rotating by $$90^\circ$$ and then reflecting in the line $$y=x$$ to the result of first reflecting in $$y=x$$ and then rotating $$90^\circ\text{.}$$

#### (c)

Find the matrix that results from composing a $$90^\circ$$ rotation with itself. Explain the geometric meaning of this operation.

#### (d)

Find the matrix that results from composing a $$90^\circ$$ rotation with itself four times; that is, if $$T$$ is the matrix transformation that rotates vectors by $$90^\circ\text{,}$$ find the matrix for $$T\circ T\circ T \circ T\text{.}$$ Explain why your result makes sense geometrically.

### Activity2.6.0.4.

Explain why the matrix that rotates vectors counterclockwise by an angle $$\theta$$ is
\begin{equation*} \left[\begin{array}{rr} \cos\theta \amp -\sin\theta \\ \sin\theta \amp \cos\theta \\ \end{array}\right]\text{.} \end{equation*}

### Activity2.6.0.5.

For this last activity, do Activity 2.6.4 6  in Understanding Linear Algebra.
davidaustinm.github.io/ula/sec-transforms-geom.html#activity-linear-trans-geom
Activity 2.6.4 davidaustinm.github.io/ula/sec-transforms-geom.html#activity-26