Orthogonal Sets to Find Coordinates

Corrolary

1. If $\{{\alpha }_1,\dots ,{\alpha }_m\}$ is a Orthogonal Set of non-zero vectors that spans $V$
2. Then, $\forall \beta \in V,\beta ={\sum }_{i=1}^m\frac{⟨\beta |{\alpha }_k⟩}{||{\alpha }_k|{|}^2}{\alpha }_k$

This allows you to find a coordinate by just using the inner product with one orthogonal vector. This ties into how nice matrixes are formed.

Example

• Find the coordinate form of $(1,3)$ using the orthogonal basis $\beta =\{\frac{1}{\sqrt{2}}(1,1),\frac{\sqrt{1}}{2}(1,-1)\}$

[(1,3)] = [\begin{array}{cc} \langle (1,3) | (\frac{1}{\sqrt{ 2 }}, \frac{1}{\sqrt{ 2 }}) \\ \langle (1,3) | (\frac{1}{\sqrt{ 2 }} , - \frac{1}{\sqrt{ 2 }})\\ \end{array}]$$ $$[(1,3)] = (\frac{4}{\sqrt{ 2 }}, -\frac{2}{\sqrt{ 2 }})$$