###### Exercise4

We have seen how a matrix transformation can perform a geometric operation; now 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 $$90^\circ\text{.}$$ 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 $$S\circ T(\xvec) = \xvec\text{.}$$ 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 $$T^{-1}\text{.}$$

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

3. The matrix transformation $$T:\real^2\to\real^2$$ defined by the matrix

\begin{equation*} A= \left[\begin{array}{rr} 1\amp 1 \\ 0 \amp 1\\ \end{array}\right] \end{equation*}

is called a shear. Find the inverse of $$T\text{.}$$

4. Describe the geometric effect of the matrix transformation defined by

\begin{equation*} A=\left[\begin{array}{rr} \frac12 \amp 0 \\ 0 \amp 3\\ \end{array}\right] \end{equation*}

and then find its inverse.

in-context