The geometry of matrix transformations

Section 2.6 The geometry of matrix transformations

Activity 2.6.0.1.

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 \(A=\left[\begin{array}{rr} 1 \amp -1 \\ -2 \amp 2 \\ \end{array}\right]\text{.}\)

Activity 2.6.0.2.

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.

Activity 2.6.0.3.

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.

Activity 2.6.0.4.

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*}

Activity 2.6.0.5.

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

TBIL Activities for Understanding Linear Algebra

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