A First Course in Linear Algebra · Problem CB.C40

Representations — Robert Beezer

Mathematics — Linear Algebra · diagonal-matrix-representation-symmetric

Problem

Let

S_{22}

be the vector space of

2 \times 2

symmetric matrices. Find a basis

C

for

S_{22}

that yields a diagonal matrix representation of the linear transformation

R

R: S_{22} \to S_{22}, \quad R\left(\begin{bmatrix}a&b\\b&c\end{bmatrix}\right) = \begin{bmatrix}-5a+2b-3c & -12a+5b-6c\\ -12a+5b-6c & 6a-2b+4c\end{bmatrix}