  Motivated by an influential result of Bourgain and Tzafriri, we consider
continuous matrix functions $A:\mathbb{R}\to M_{n\times n}$ and lower
$\ell_2$-norm bounds associated with their restriction to certain subspaces. We
prove that for any such $A$ with unit-length columns, there exists a continuous
choice of subspaces $t\mapsto U(t)\subset \mathbb{R}^n$ such that for $v\in
U(t)$, $\|A(t)v\|\geq c\|v\|$ where $c$ is some universal constant.
Furthermore, the $U(t)$ are chosen so that their dimension satisfies a lower
bound with optimal asymptotic dependence on $n$ and $\sup_{t\in
\mathbb{R}}\|A(t)\|.$ We provide two methods. The first relies on an
orthogonality argument, while the second is probabilistic and combinatorial in
nature. The latter does not yield the optimal bound for $\dim(U(t))$ but the
$U(t)$ obtained in this way are guaranteed to have a canonical representation
as joined-together spaces spanned by subsets of the unit vector basis.
