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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?

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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

$$[T]_\beta=P^{-1}[T]_\alpha P$$

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

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