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Linear Algebra

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Moderate

COB Formula on two Bases of $R^n$

LINALG-4GUNWY

Let:

$$\mathcal{B}=\{b_1,\ldots,b_n\}$$

$$\mathcal{B}'=\{b_1',\ldots,b_n'\}$$

...be two bases of $\mathbb{R}^n$, and let $P$ be the matrix whose $i$-th column vector is the coordinate vector of $b_i$ with respect to $\mathcal{B}'$.

If:

$$T:\mathbb{R}^n\to\mathbb{R}^n$$

...is a linear transformation whose matrix with respect to the basis $\mathcal{B}$ is $M$, then what is the matrix of $T$ with respect to $\mathcal{B}'$?

A

$PMP^T$

B

$P^TMP$

C

$PMP^{-1}$

D

$P^{-1}MP$

E

$M$