Do gamma matrices form a basis? Do the four gamma matrices form a basis for the set of matrices $GL(4,\mathbb{C})$? I was actually trying to evaluate a term like $\gamma^0 M^\dagger \gamma^0$ in a representation independent way, where $M, M^\dagger$ are $4\times 4$ matrices.
 A: As previous answers have correctly noted gamma matrices do not forma a basis of $M(4,\mathbb{C})$. Nevertheless you can construct one from them in the following way


*

*1 the identity matrix $\mathbb{1}$

*4 matrices  $\gamma^\mu$

*6 matrices  $\sigma^{\mu\nu}=\gamma^{[\mu}\gamma^{\nu]}$

*4 matrices  $\sigma^{\mu\nu\rho}=\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho]}$

*1 matrix    $\sigma^{\mu\nu\rho\delta}=\gamma^{[\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\delta]}=i\epsilon^{\mu\nu\rho\delta}\gamma^5$
these 16 matrices form the basis that we were looking for.
Furthermore they are used to construct the spinor bilinears multiplying by $\bar{\psi}$ on the left and $\psi$ on the right, which transform in the Lorentz indices as follows


*

*$\bar{\psi}\psi$ scalar

*$\bar{\psi}\gamma^\mu\psi$ vector

*$\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor

*$\bar{\psi}\sigma^{\mu\nu\rho}\psi$ pseudovector 

*$\bar{\psi}\gamma^5\psi$ pseudoscalar


the fact that they form a basis of of $M(4,\mathbb{C})$ is very important because these are then the only independent spinor bilinears (i.e $\bar{\psi}M\psi$) that can be constructed, any other can be expressed a linear combination of these. A different issue is if it would make any sense to sum any of these since they are different types of tensors under Lorentz group transformations.
A: To complement V. Moretti's excellent answer, it's worth emphasizing that the dimension of the four-by-four complex matrices $\mathbb C^{4\times 4}$, when seen as a vector space over $\mathbb C$, is $4\!\times\!4=16$. As such, a set of four matrices (i.e. vectors in $\mathbb C^{4\times 4}$) can never be a basis for it.
It's also worth saying that the general linear group $\text{GL}(4,\mathbb C)\subset\mathbb C^{4\times 4}$, i.e. the four-by-four matrices with nonzero determinant, is not a vector space (for one, it doesn't have a zero), and it is therefore misleading to speak of a basis for it. That said, it is still possible to ask for a minimal set of vectors (i.e. matrices) whose span will contain $\text{GL}(4,\mathbb C)$; this turns out to require a full basis of $\mathbb C^{4\times 4}$ because the matrices you 'skip', $\mathbb C^{4\times 4}\setminus\text{GL}(4,\mathbb C)$, have measure zero, so $\text{GL}(4,\mathbb C)$ is a complex manifold of dimension 16 and requires that many parameters to describe.
A: No they do not, due to dimensional reasons, but they are generators of the algebra. That is, $I$ and the products of $\gamma^a$ (products of one, two, three and four matrices) form such a basis.
NOTE ADDED. As Emilio Pisanty correctly remarked (also making some further interesting comments) $GL(4, \mathbb C)$ is not a linear space so questions about bases of it are inappropriate. In fact I implicitly interpreted that $GL(4,\mathbb C)$ as $M(4, \mathbb C)$, the complex algebra of $4\times 4$ complex matrices which, by definition,  is also a complex vector space.
