Math Help - Explaining why a set of orthonormal vectors spans a subspace

Suppose we have a subspace V spanned by the basis {1, x, x^2, x^3, sin(x)}.


By applying the Gram-Schmidt process to the above basis, we obtain an orthonormal basis {z1, z2, z3, z4, z5}.


Why does this orthonormal basis span the same subspace V (that was spanned by the non-orthonormal vectors)?

Because the orthonormal basis members are linear combinations of the original vectors, so by adjusting coefficients you can span the same subspace.

I feel like this was a very short explanation, someone could probably provide a better one

I think it was very nicely stated!

I think it's starting to make sense, thank you! Would you mind explaining how we can determine the weights to express the orthonormal vectors as linear combinations of the original vectors?