Chapter 7.CM, Problem 18CM

To determine

(a)

The matrix $A$ of given linear transformation relative to the basis $B$.

To determine

(b)

The transition matrix $P$ from $B^{\prime }$ to $B$.

To determine

(c)

The matrix $A^{\prime }$ of the given linear transformation relative to the basis $B^{\prime }$.

To determine

(d)

The value of ${\left[T\left(v\right)\right]}_{B^{\prime }}$ when ${\left[v\right]}_{B^{\prime }}=\left[\begin{array}{c}3\\ -2\end{array}\right]$.

To determine

(e)

To verify:

The answer in part (d) by finding ${\left[v\right]}_B$ and ${\left[T\left(v\right)\right]}_B$.