16 - Gram-Schmidt Orthonormalisation Process

Gram-Schmidt Orthonormalisation Process

Let $\mathcal{B}=\{v_1,\cdots ,v_n\}$ be a basis of the inner product space $V$ over $\mathbb{K}=\mathbb{R},\mathbb{C}$
Define

$$\begin{array}{rl}{w}_1& =v_1\\ w_2& =v_2-\frac{⟨w_1,v_2⟩}{⟨w_1,w_1⟩}{w}_1\\ ⋮& \\ w_k& =v_k-\sum _{i=1}^{k-1}\frac{⟨w_i,v_k⟩}{⟨w_i,w_i⟩}{w}_i\\ ⋮& \end{array}$$

Assuming that $⟨w_1,cdost,w_{k-1}⟩=⟨v_1,\cdots ,v_{k-1}⟩$, the general form from above shows that

$$⟨w_1,\cdots ,w_k⟩=⟨w_1,\cdots w_{k-1},v_k⟩=\cdots =⟨v_1,\cdots ,v_k⟩$$

Assuming that $\{w_1,\cdots ,w_{k-1}\}$ are orthogonal then for $j<k$

$$⟨w_j,w_k⟩=⟨w_j,v_k⟩-\frac{⟨w_j,v_k⟩}{⟨w_j,w_j⟩}⟨w_j,w_j⟩=0$$

So by induction $D=\{w_1,\cdots ,w_n\}$ is orthogonal, spanning
Hence by Linear Independence of an Orthogonal Set then

$$\mathbf{D}\text{ is an orthogonal basis}$$

Define

$$u_i=\frac{w_i}{||w_i||}\text{ where }||w_i||=\sqrt{⟨w_i,w_i⟩}$$

Then

$$\mathcal{E}=\{u_i,\cdots ,u_n\}\text{ is a orthonormal basis }$$