Chapter 6.5, Problem 16P

let $T_1$ be the linear transformation from Problem 13 and let $T_2$ be the linear transformation from Problem 15

(a) Find the matrix representation of $T_2{T}_1$ relative to the standard bases.

(b) Verify Theorem 6.5.7 by comparing part (a) with the product of the matrices in Problems 13 and 15.

(c) Use the matrix representation found in (a) to determine $\left(T_2{T}_1\right)\left(2+5x-x^2+3x^4\right)$. Verify your answer by computing this directly.

13. $T:P_4\left(ℝ\right)\to P_3\left(ℝ\right)$ via $T\left(p\left(x\right)\right)=p^{\prime }\left(x\right)$, relative to the standard bases $B$ and $C$; $p\left(x\right)=3-4x+6x^2+6x^3-2x^4$.

15. $T:P_3\left(ℝ\right)\to ℝ$ via $T\left(p\left(x\right)\right)=p\left(2\right)$, relative to the standard bases $B$ and $C$; $p\left(x\right)=2x-3x^2$.

Theorem 6.5.7 If U, V, and W are vector spaces with ordered bases A, B, and C, respectively, and $T_1:U\to V$ and $T_2:V\to W$ are linear transformations, then we have

${\left[T_2{T}_1\right]}_A^C={\left[T_2\right]}_B^C{\left[T_1\right]}_A^B$ (6.5.2)