A First Course in Linear Algebra · Problem MR.C22

Representations — Robert Beezer

Mathematics — Linear Algebra · matrix-representations-symmetric-matrices

Problem

Let

S_{22}

be the vector space of

2 \times 2

symmetric matrices. Build the matrix representation of the linear transformation

T: P_2 \to S_{22}

relative to the bases

B

and

C

and then use this matrix representation to compute

T\left(3 + 5x - 2x^2\right)

B = \left\{1,\, 1+x,\, 1+x+x^2\right\}

C = \left\{\begin{bmatrix}1&0\\0&0\end{bmatrix}, \begin{bmatrix}0&1\\1&0\end{bmatrix}, \begin{bmatrix}0&0\\0&1\end{bmatrix}\right\}

T\left(a + bx + cx^2\right) = \begin{bmatrix}2a-b+c & a+3b-c\\ a+3b-c & a-c\end{bmatrix}