Rows of Change of Basis Matrix

Maybe a stupid question, but since I don't find a confirmation to my doubt on the Internet, I'll also ask here.

To change basis from A to B we use a matrix whose columns are the basis vectors of A expressed in the new basis B. But we can also say that its rows are the basis vectors of B expressed in the old basis A, can't we?

No. That's not in general true. If it were, then the change-of-basis matrix from $B$ to $A$ would be the transpose of the c-o-b matrix from $A$ to $B$. It's not, in general. Instead, it's the inverse of that matrix.
• Close. It's only true when the change-of-basis is an orthonormal matrix. That can happen even with non-orthogonal bases (example: if $A$ and $B$ are the same basis, then the change-of-basis matrix is the identity, regardless of whether they're orthogonal bases). Answer edited accordingly. – John Hughes Nov 26 '13 at 17:57