Can Gram-Schmidt algorithm be applied to every set of vectors in ${\rm I\!R}^{n}$?

Question

I have the following question: "Can Gram-Schmidt algorithm be applied to every set of vectors in ${\rm I\!R}^{n}$?".
I know that in the general case we apply it to the finite independent set of the vectors.
But can we do that with the infinite set or dependent?

Answer 1

If the vectors are not linearly independent (which is automatically the case when there are more than $n$ vectors), at some step the algorithm asks you to divide by $0$. Indeed, the algorithm "makes" each input vector orthogonal to all previous ones by translating it parallel to the span of the previous ones. If the vectors is already in the subspace spanned by the previous vectors, this orthogonal vector must be the zero vector.