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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?

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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.

If the change of basis matrix is orthogonal, then the inverse is the transpose, in which case your statement about the rows is correct.

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  • $\begingroup$ So we can say it's true only with basis systems constituted by orthogonal vectors, right? $\endgroup$
    – Pippo
    Nov 26, 2013 at 17:52
  • $\begingroup$ 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. $\endgroup$
    – John Hughes
    Nov 26, 2013 at 17:57

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