# Change of basis matrix verification

Let $B$ and $C$ be two bases. To find the change of basis matrix $\phi_{B,C}$, I compute $\phi_{SB,B}$ and $\phi_{SB,C}$. Create the new matrix $T=[\phi_{SB,B}|\phi_{SB,C}]$. Reducing it to reduced row echelon form, should yield $T=[I|\phi_{B,C}$, right? Is there a way to verify that the new basis indeed maps $[v]_B$ to $[v]_C$?

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What are $\phi_{SB, B}$ and $\phi_{SB, C}$? – student Jan 19 '12 at 13:21
@Leandro The change of basis from the standard basis to B, respectively C. – Andrew Jan 19 '12 at 13:23

To check that the change of basis is correct, we can multiply $B$ times $\phi_{B,C}$ and we should get the $C$ basis. This can tell you that $\phi_{B,C}$ is correct.
The matrix $\phi_{B,C}$ should have columns that are the coordinates of the $B$-basis vectors in terms of the $C$-basis vectors (i.e., old basis in terms of new basis). Let your matrix $T$ have left block whose columns are the $C$ basis vectors, and right block whose columns are the $B$-basis vectors. Row reduce $T$. The right hand block will be $\phi_{B,C}$.