The geometry of matrix transformations

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

1. Find the matrix of the transformation that has no effect on vectors; that is, $T(\xvec) = \xvec$.

2. Find the matrix of the transformation that reflects vectors in $\real^2$ across the line $y=x$.

3. What is the result of composing the reflectio…

For each of the following geometric operations in the plane, find a $2\times 2$ matrix that defines the matrix transformation performing the operation.

1. Rotates vectors by $180^\circ$.

2. Reflects vectors across the vertical axis.

3. Reflects vectors across the line $y=-x$.

4. Rotates vectors counterclockwise by $60^\circ$.

5. First rotates vectors counterclockwise by $60^\circ$ and then reflects in the line $y=x$.

This exercise investigates the composition of reflections in the plane.

1. Find the result of first reflecting across the line $y=0$ and then $y=x$. What familiar operation is the cumulative effect of this composition?

2. What happens if you compose the operations in the opposite order; that is, what happens if you first reflect across $y=x$ and then $y=0$? What familiar operation results?

3. What familiar geometric operatio…

If a matrix transformation performs a geometric operation, we would like to find a matrix transformation that undoes that operation.

1. Suppose that $T:\real^2\to\real^2$ is the matrix transformation that rotates vectors by $90^\circ$. Find a matrix transformation $S:\real^2\to\real^2$ that undoes the rotation; that is, $S$ takes $T(\xvec)$ back into $\xvec$ so that $(S\circ T)(\xvec) = \xvec$. Think geometrically about what the transformation $S$ should be and then verify it algebraically.

   We…