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
\(\seteqnumber{0}{22.}{164}\)
\begin{equation} x = \sum _{1 \leqq k \leqq n} \frac {\langle x, v_k \rangle }{||v_k||^2} v_k \end{equation}

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.

\begin{equation} x = \sum _{1 \leqq j \leqq n} a_i v_i \end{equation}

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