Proof of Trace Orthogonality Relation for Matrices $\Gamma^A$

[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?

 

[Fredrik]

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.