# A calculus of the absurd

##### 22.7.5 Orthonormal bases
• Theorem 22.7.2 Let $$V$$ be a finite-dimensional vector space, and let $$\beta = \{v_1, v_2, ..., v_n\}$$ be a basis for this vector space. Then, (spoiler alert) we know that for all $$x \in V$$ that

$$x = \sum _{1 \leqq k \leqq n} \frac {\langle x, v_k \rangle }{||v_k||^2} v_k$$

We can prove this as follows. First we know that for some scalars $$a_1, a_2, ..., a_n$$ that as $$\beta$$ is a basis.

$$x = \sum _{1 \leqq j \leqq n} a_i v_i$$

We now want to find the values of $$a_i$$ (for any $$i$$). We know that for all $$j$$

\begin{align} \langle x, v_j \rangle &= \left \langle \sum _{1 \leqq j \leqq n} a_i v_i, v_j \right \rangle \\ &= \sum _{1 \leqq j \leqq n} a_i \langle v_i, v_j \rangle \end{align}

Then as $$\beta$$ is an orthogonal basis, we know that all the $$\langle v_i, v_j \rangle$$ terms are zero