Can $|Mu|$ be arbitrary close to 0 for a non-singular $M$ and a unit $u$?

Question

Let M be a $r \times c$ matrix with entries in $\mathbb R$ that has a left inverse.

Does there exist $\epsilon > 0$ such that $|Mu| > \epsilon$ for every vector $u \in R^{c}$ of length $1$?

Or can one find a sequence of unit vectors ${u_{n}}$ such that $|Mu_{n}| \to 0$?

Answer 1

If the sequence $Mu_{n}$ converges to $0$, then the sequence $u_{n} = M^{-1}Mu_{n}$ converges to $M^{-1}0 = 0$ (because the left inverse $M^{-1}$ exists and the function $x \to M^{-1}x$ is continuous). However, each $u_n$ is a unit vector, so $|u_{n}| \to 1$ and the sequence can not converge to $0$.