# Theorem about coordinates of a subspace in an inner product space

Theorem: Let $$B = {v_1,v_2,...,v_n}$$ be an orthonormal basis of an inner product space V and let $$w∈V$$.Then the coordinates $$a_1,a_2,...,a_n$$ of $$w$$ in the basis B are given by $$a_i =⟨w,v_i⟩$$.

Do coordinates in this case mean scalars on subspace w?

• Which subspace is that? Mar 7 '20 at 22:20
• @JoséCarlosSantos inner product space Mar 7 '20 at 23:11

Given a basis $$\mathcal{B}=(v_1,\ldots, v_n)$$ for a vector space $$V$$ over a field $$k$$ $$(=\mathbb{R},\mathbb{C}$$ for instance), and given an element $$w\in V$$, we say that the coordinates for $$w$$ with respect to $$\mathcal{B}$$ are the unique $$a_1,\ldots, a_n\in k$$ such that $$w=a_1v_1+\cdots+a_nv_n$$.
Of course, the facts that such $$a_1,\ldots, a_n$$ exist and are unique follow from the definition of a basis.