# Space time algebra isomorphic to matrix algebra

i have the following problem:

I already know that there exists representation of the Clifford Algebra of the Minkowski space $\mathcal{C}l(M,\eta)$. Here $M$ denotes the Minkowski space and $\eta$ is the metric tensor with the signatur $(+1,-1,-1,-1)$.

I know that a representation is given by by the map $\gamma:M\to M_4(\mathbb{C})$ by $\gamma(e_\mu)=\gamma_\mu$, where $\gamma_\mu$ denotes the gamma matrix.

Now I want to show, that the spaces are isomorphic. I've already shown the defining relation $\gamma_\mu\gamma_\nu+\gamma_\nu\gamma_\mu=2\eta_{\mu\nu}$.

My problem is to show that the defined mapping $\gamma$ is bijective. An advice was given in the form that it is enough to show that the mapping is injective or surjective because of dimension of the spaces.

The gamma matrices are four $\Bbb C$-linearly independent matrices of $M_2(\Bbb C)$, a space with $\Bbb C$-dimension $4$, so the gamma matrices form a basis. Since the image of $\gamma$ contains this basis, it is the entire space.
Thus $\gamma$ is surjective and hence injective.