# Basis expansion

1. Oct 29, 2014

### gentsagree

I know that the matrices {$\Gamma^{A}$} obey the trace orthogonality relation $Tr(\Gamma^{A}\Gamma_{B})=2^{m}\delta^{A}_{B}$

In order to show that a matrix M can be expanded in the basis $\Gamma^{A}$ in the following way

$$M=\sum_{A}m_{A}\Gamma^{A}$$
$$m_{A}=\frac{1}{2^{m}}Tr(M\Gamma_{A})$$

is it enough to just substitute the first equation for M in the second, and work out that the RHS is indeed equal to $m_{A}$ (using the orthogonality), or is this just a mere verification, and not a proof?

2. Oct 29, 2014

### Fredrik

Staff Emeritus
What you're proving there is that if a vector $v$ is in the subspace spanned by an orthonormal set $\{e_i\}$, then $v=\sum_i \langle e_i,v\rangle e_i$.

To prove that an arbitrary $v$ in your vector space is in the subspace spanned by that orthonormal set, you will have to do something else. You will have to write down a definition of that specific vector space, and then use it somehow.