Change of basis formula

Statement

Let $F$ be a field and let $V$ and $W$ be finite-dimensional vector spaces over $F$. Let $B, B'$ be bases of $V$, and let $C, C'$ be bases of $W$. Let $P$ be the change of basis matrix from $B$ to $B'$, and $Q$ the change of basis matrix from $C$ to $C'$. Then, for each linear map $T \colon V \to W$, $$[T]_{C',B'} = Q^{-1} [T]_{C,B} P,$$ where $[T]$ denotes the matrix of $T$ in the given bases. In particular, for an endomorphism $T$ of $V$ and $C = B$, $C' = B'$, one gets $[T]_{B'} = P^{-1} [T]_B P$.

Remarks

The formula follows from the matrix of a composition, applied to $T = \id_W \circ T \circ \id_V$. For endomorphisms, the two matrices are similar. For linear maps between two spaces, the two matrices are equivalent.

Explanations

L1 human

When we change bases, the matrix of a linear map changes in a fixed way. We translate the input from the new to the old coordinates, apply the old matrix, and translate the result back. This gives the product $Q^{-1} A P$.