  We consider sample covariance matrices of the form $X^*X$, where $X$ is an $M
\times N$ matrix with independent random entries. We prove the isotropic local
Marchenko-Pastur law, i.e. we prove that the resolvent $(X^* X - z)^{-1}$
converges to a multiple of the identity in the sense of quadratic forms. More
precisely, we establish sharp high-probability bounds on the quantity $\langle
v, (X^* X - z)^{-1} w \rangle - \langle v,w\rangle m(z)$, where $m$ is the
Stieltjes transform of the Marchenko-Pastur law and $v, w \in \mathbb C^N$. We
require the logarithms of the dimensions $M$ and $N$ to be comparable. Our
result holds down to scales $Im z \geq N^{-1+\epsilon}$ and throughout the
entire spectrum away from 0. We also prove analogous results for generalized
Wigner matrices.
