Coordinatization over orthogonal vectors
Dependencies:
Let $V$ be an inner product space. Let $X = \{x_1, x_2, \ldots, x_n\}$ be a set of orthogonal vectors in $V$. Then for every $v \in \operatorname{span}(X)$,
\[ v = \sum_{i=1}^n \frac{\langle v, x_i \rangle}{\|x_i\|^2} x_i \]
Proof
\begin{align} v &= \sum_{i=1}^n a_ix_i \tag{$\because v \in \operatorname{span}(X)$} \\ \Rightarrow \langle v, x_k \rangle &= \left\langle \sum_{i=1}^n a_ix_i , x_k \right\rangle \\ &= \sum_{i=1}^n a_i \langle x_i, x_k \rangle \tag{linearity in first argument} \\ &= a_k \langle x_k, x_k \rangle = a_k \|x_k\|^2 \tag{$\because \forall i \neq k, \langle x_i, x_k \rangle = 0$} \\ \Rightarrow a_k &= \frac{\langle v, x_k \rangle}{\langle x_k, x_k \rangle} \tag{$x_k \neq 0 \Rightarrow \|x_k\|^2 \neq 0$} \end{align}
Dependency for: None
Info:
- Depth: 6
- Number of transitive dependencies: 6