0
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ So we can say it's true only with basis systems constituted by orthogonal vectors, right? $\endgroup$ – Pippo Nov 26 '13 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 '13 at 17:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.