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If $P$ is the transition matrix from a basis $B'$ to a basis $B$, and $Q$ is the transition matrix from $B$ to a basis $C$, what is the transition matrix from $B'$ to $C$? What is the transition matrix from $C$ to $B'$?

I'm not sure how to approach this problem, thanks for helping :)

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Imagine you begin with a vector $\vec{v}_{B}'$ in basis $B'$, in order to represent it in the basis $B$, we have:


Now transforming from the basis $B$ to the basis $C$ we get:


And therefore the transition matrix from $B'$ to $C$ is $\mathbf{QP}$.

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I have always found this notation helpful: let $$[I]_\alpha^\beta$$ denote the change-of-basis matrix from the basis $\alpha$ to $\beta$. Then to go from $\alpha$ to $\gamma$ through $\beta$ in between we have $$ [I]_\alpha^\gamma = [I]_\beta^\gamma [I]_\alpha^\beta.$$ So if you ever get mixed up, just read the matrices right to left, matching upper right with lower left, and with the proper origin and target bases. The matrices in your notation converted to my notation would be $\mathbf{P} = [I]_{B'}^B$ and $\mathbf{Q} = [I]_B^C$, so the change of basis matrix from $B'$ to $C$ is $$[I]_{B'}^C = [I]_B^C[I]_{B'}^B = \mathbf{Q}\mathbf{P}.$$ To go from $C$ to $B'$ we simply invert the matrix $[I]_{B'}^C$, so the resulting change of basis matrix is $$[I]_C^{B'} = ([I]_{B'}^C)^{-1} = (\mathbf{Q}\mathbf{P})^{-1} = \mathbf{P}^{-1}\mathbf{Q}^{-1}.$$

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