Exercise 4

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 say that $S$ is the inverse of $T$ and we will write it as $T^{-1}$.

2. Verify algebraically that the reflection $R:\real^2\to\real^2$ across the line $y=x$ is its own inverse; that is, $R^{-1} = R$.

3. The matrix transformation $T:\real^2\to\real^2$ defined by the matrix $$A= \left[\begin{array}{rr} 1& 1 \\ 0 & 1\\ \end{array}\right]$$ is called a shear. Find the inverse of $T$.

4. Describe the geometric effect of the matrix transformation defined by $$A=\left[\begin{array}{rr} \frac12 & 0 \\ 0 & 3\\ \end{array}\right]$$ and then find its inverse.