Find a matrix representation of the linear transformation T relative to the bases B and C.
\(T:P_{ 2 }\rightarrow C^{ 2 },T(p\left( x \right) )=\begin{bmatrix} p(1) \\ p(3) \end{bmatrix}\)
\(B = \left\{ 2 - 5x + x^ 2, 1 + x - x^ 2, x^ 2 \right\}\)
\(C = \left\{ \begin{bmatrix} 3 \\ 4 \end{bmatrix}, \begin{bmatrix} 2 \\ 3 \end{bmatrix}\right\}\)
Solution:
By applying definiton MR located on page 528 the matrix representation is the mxn matrix \({ M }^{ T }_{ B,C } = pc(T(2 - 5x + x^2))|pc(T(1 + x - x^2))|pc(T(x^2))\)
\(pc(T(2 - 5x + x^2))\)
= \(pc(\begin{bmatrix} -2 \\ -4 \end{bmatrix})\)
= \(pc(2 \begin{bmatrix} 3 \\ 4 \end{bmatrix} - 4 \begin{bmatrix} 2 \\ 3 \end{bmatrix})\)
= \(\begin{bmatrix} 2 \\ -4 \end{bmatrix}\)
\(pc(T(1 + x - x^2))\)
= \(pc(\begin{bmatrix} 1 \\ -5 \end{bmatrix})\)
= \(pc(13 \begin{bmatrix} 3 \\ 4 \end{bmatrix} - 19 \begin{bmatrix} 2 \\ 3 \end{bmatrix})\)
= \(\begin{bmatrix} 13 \\ -19 \end{bmatrix}\)
\(pc(T(x^2))\)
= \(pc(\begin{bmatrix} 1 \\ 9 \end{bmatrix})\)
= \(pc(-15 \begin{bmatrix} 3 \\ 4 \end{bmatrix} + 23 \begin{bmatrix} 2 \\ 3 \end{bmatrix})\)
= \(\begin{bmatrix} -15 \\ 23 \end{bmatrix}\)
The matrix representation is \({ M }^{ T }_{ B,C } = \begin{bmatrix} 2 & 13 & -15 \\ -4 & -19 & 23 \end{bmatrix}\)