# Change of basis of linear map

Suppose T: $\mathbb{R}^{2}$$\rightarrow\mathbb{R}$$^{2}$ is linear and has matrix $\begin{pmatrix}4&9\\1&1\end{pmatrix}$ with

respect to the standard basis of $\mathbb{R}$$^{2}. What is the matrix of T with respect to the basis \beta= {(1,-1), (-3,2)} ?. How do we approach these sort of problems, commutative diagram? And if so how would it look? ## 1 Answer The transition matrix from the standard basis \alpha to the basis \beta is:$$P=\begin{pmatrix}1&-3\\-1&2\end{pmatrix}$$so the matrix of T relative to \beta is$$[T]_\beta=P^{-1}[T]_\alpha P$\$

• I hope you're having a great day! – Namaste Jul 31 '14 at 11:24