1. Why Orthogonal Bases Are Worth Manufacturing
The Big Idea
In a general basis, finding the coordinates of a vector requires solving a linear system. In an orthogonal basis, you can simply read off each coordinate using a dot product — no row reduction needed.
Coordinates the Hard Way (General Basis)
Suppose you have a basis $\{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}$ for a subspace $W$, and you want to express a vector $\mathbf{x} \in W$ as:
$$\mathbf{x} = c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_n \mathbf{v}_n$$You must solve a system of $n$ equations in $n$ unknowns — set up the augmented matrix $[\mathbf{v}_1 \;|\; \mathbf{v}_2 \;|\; \cdots \;|\; \mathbf{v}_n \;|\; \mathbf{x}]$ and row reduce. This is $O(n^3)$ work.
Coordinates the Easy Way (Orthogonal Basis)
If instead $\{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n\}$ is an orthogonal basis (meaning $\mathbf{u}_i \cdot \mathbf{u}_j = 0$ whenever $i \neq j$), then the expansion coefficient for each basis vector is simply:
Why does this work? Take $\mathbf{x} = c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \cdots + c_n\mathbf{u}_n$ and dot both sides with $\mathbf{u}_i$:
$$\langle \mathbf{x}, \mathbf{u}_i \rangle = c_1\underbrace{\langle \mathbf{u}_1, \mathbf{u}_i \rangle}_{0} + \cdots + c_i\langle \mathbf{u}_i, \mathbf{u}_i \rangle + \cdots + c_n\underbrace{\langle \mathbf{u}_n, \mathbf{u}_i \rangle}_{0} = c_i \langle \mathbf{u}_i, \mathbf{u}_i \rangle$$