I have a matrix $M \in \mathbb{R}^{n\times n}$ such that
$$ M = EE^\top $$ where $E \in \mathbb{R}^{n\times k}$ is full rank.
$M$ can also be expressed in terms of a Singular value decomposition:
$$ M = USU^\top $$
Can I say that the vector space generated by the columns of $U$ is the same of the one generated by the columns of $E$?
I assume that in your version of SVD, $U$ is $n \times k$ and $S$ is $k \times k$.
Indeed, the column space of $U$ is identical to the column space of $E$.