Problem with subspace projections

Question

I have been looking at some exercises about vector x being projected onto subspaces but I have a question. In the exercises I solved, the subspace onto which the vector x is projected happens to be horizontal. I understand that the basic procedure is to decompose x in a vertical and horizontal component. In the exercise forementioned they logically claim the horizontal component coincides with the projection of a x onto the subspace because our subspace is horizontal. However, what happens if the subspace formed by the vectors is not horizontal?

Answer 1

If the subspace in question has an orthonormal basis $v_1,v_2,\dots,v_n$ then the projection of $v$ onto that subspace is $(v\cdot v_1)v_1+(v\cdot v_2)v_2+\cdots+(v\cdot v_n)v_n$.

If you don't have an orthonormal basis for the subspace, life is more complicated.