📐 Change of Basis Formula

If $[T]_\\beta$ and $[T]_{\\beta\

Proof: Let $I: V \\to V$ be the identity map. Then $[I]_{\\beta'}^\\beta = P$ is the change of basis matrix.

We have $T = I \\circ T \\circ I$, so:

$$[T]_{\\beta'} = [I]_\\beta^{\\beta'} [T]_\\beta [I]_{\\beta'}^\\beta = P^{-1} [T]_\\beta P$$

From: Advanced Linear Algebra

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