Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$

Question

Does a orthogonal basis for the span of $S$ always have the same dimension as the basis of $S$

Basically if I have found the orthonormal basis for the span of S can I use that to find the dimension of the span of $S$. Or do I need to do a separate calculation... row reduction etc.?

Answer 1

Let's say we have an orthonormal basis for $S$. As long as you are definitely sure that the basis really is a basis for $S$, then yes, that orthonormal basis has the same number of vectors as any other valid basis in $S$.

Answer 2

For infinite-dimensional Hilbert spaces, the term "basis" can have two distinct meanings. In the context of an orthonormal basis, infinite sums are allowed. However, in the context of a vector space basis (sometimes called a Hamel basis), only finite sums can be considered. Thus for an infinite-dimensional Hilbert space, an orthonormal basis is not a vector space basis. The cardinality of an orthonormal basis can differ from the cardinality of a vector space basis.

For example, the Hilbert space $\ell^2$ of square-summable sequences has a countable orthonormal basis, but every vector space basis of $\ell^2$ is uncountable.