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Given a basis $B$ (of a vector space $V$), and a transition matrix $P_{B\to B'}$, is it possible to find the basis $B'$?

I'm working through change of basis, and I can't work out how (or find any information online) on how, or even it's possible.

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The transition matrix is the matrix whose columns are the coordinates of the vectors of $B'$ with respect to the basis $B$. –  1015 Feb 7 '13 at 15:09

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up vote 1 down vote accepted

I will assume first that $P_{B\rightarrow B'}$ means the matrix of: $$ \mbox{id}:(V,B)\longrightarrow (V,B'). $$ Then $Q=P_{B\rightarrow B'}^{-1}$ is the matrix of: $$ \mbox{id}:(V,B')\longrightarrow (V,B). $$ So the columns of $Q$ are the coordinates of the vectors of $B'$ with respect to the basis $B$. Knowing the vectors of $B$, you can now compute those of $B'$.

But if $P_{B\rightarrow B'}$ means the matrix of: $$ \mbox{id}:(V,B)\longrightarrow (V,B') $$ then its columns are already the coordinates of the vectors of $B'$ with respect to $B$ etc..., so no need to invert it.

Note: I believe the first interpretation is the most universal one.

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