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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