Linear independence and orthogonaliy

[Werg22]

Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

 

[A KillEase]

If it's in an inner product space, you can use Gram-Schmidt to find T.

 

[intrepid_nerd]

Please refresh my memory; if a finite set S is L.I., then does this imply the existence of a set T of the same size (i.e. |T| = |S|) so that the elements T are pairwise orthogonal?

yes. i believe any method you use (eigenspace, nullspace, etc) to find a set that spans T will be mutually orthogonal. i'd probably guess that the basis spanning T would be orthogonal to the basis spanning A as well.