Find a matrix representation of the linear transformation T relative to the bases B and C.
\(T : P2 \quad to\quad C2, T (p(x)) = \begin{bmatrix}p(1)\\p(3)\end{bmatrix}\)
\(B=\{2-5x+{x}^{2},1+x-{x}^{2},{x}^{2}\}\)
We have to evaluate the transformation at each member of the Basis B, then determine the vector that will represent the result in C.
Starting with the first term:
\(C=\{ \begin{bmatrix} 3 \\ 4 \end{bmatrix},\quad \begin{bmatrix} 2 \\ 3 \end{bmatrix}\}\)
\(2-5x+{ x }^{ 2 }\):
\(\begin{bmatrix} -2 \\ -4 \end{bmatrix}\quad\)
\({ \rho }_{ c }=\)
A <- matrix(c(3,4,2,3), 2,2)
b <- c(-2,-4)
solve(A,b)## [1] 2 -4\(1+x-{ x }^{ 2 }\):
\(\begin{bmatrix} 1 \\ -5 \end{bmatrix}\quad\) \({ \rho }_{ c }=\)
b <- c(1,-5)
solve(A,b)## [1] 13 -19\({ x }^{ 2 }\):
\(\begin{bmatrix} 1 \\ 9 \end{bmatrix}\quad\)
\({ \rho }_{ c }=\)
b <- c(1,9)
solve(A,b)## [1] -15 23Collecting all the rhos:
\({ M }_{ b,c }^{ T }=\begin{bmatrix} 2 & 13 & -15 \\ -4 & -19 & 23 \end{bmatrix}\)