Change of Basis with Orthonormal Basis

When given a orthonormal basis, you can convert a Coordinate Vector into the orthonormal basis representation using Standard Inner Product.

Example

1. Compute coordinate vector for $(1,3)$ using basis $\beta =\left\{\frac{1}{\sqrt{2}}(1,1),\frac{1}{\sqrt{2}}(1,-1)\right\}$

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