A calculus of the absurd

22.7.6 Gram-Schmidt orthonormalisation

Gram-Schmidt orthonormalisation provides a useful way to turn a set of linearly independent vectors into a set of orthogonal vectors.

The key idea is that we build our set of vectors inductively, i.e. if our set of vectors is $S$ (a finite subset of some vector space $V$). Then we will order our set (doesn’t matter how, any ordering will do) and start to build sets $S'_1$, $S'_2, ..., S'_{|S|}$ such that $S'_k$ is an orthogonal set of $k$ vectors which satisfies

\begin{equation} \Span (\text {first k vectors in $S$}) = \Span (S'_k). \end{equation}

Clearly the main thing which is missing here is the step which takes us from $S'_{k}$ to $S'_{k+1}$. There are a lot of ways to find this step,

• Consider specific examples of linear vectors in well-known vector spaces (for example $\mathbb {R}^2$) and guess the formula ¹³⁶¹³⁶ Pun entirely unintended. for performing this orthonormalisation process.
• Try to write a proof for our method and through this try to fill in the actual method.

I will try for the latter, because I think it is an approach which is much more fun. We will start by creating $S'_1$ by simply selecting the only element in the first one element of $S$ which is by itself orthogonal.

Now, suppose that we have constructed $S'_k$. We would like to find a way to build $S'_{k+1}$. Clearly we should add $s_{k+1}$ to this set, the question of course is how. We need our new vector, say $s'_{k+1}$ to be such that

\begin{align} \langle s'_{k+1}, s'_j \rangle = 0 & \forall j, 1 \leqq j \leqq k \end{align}