Is $XX^T$ invertible?

Question

In one of the lectures today, the professor said that if $X \in \mathbb{R}^{m \times n}$ matrix, and the columns of $X$ span $\mathbb{R}^m$, then the matrix $XX^T$ is invertible. I am not sure why this is the case unless $m=n$?

Answer 1

If $XX^\top v = 0$, then $\|X^\top v\|^2=v^\top XX^\top v=0$, and thus $X^\top v = 0$. Since the rows of $X^\top$ span $\mathbb{R}^m$, we must have $v = 0$, and thus the kernel of $XX^\top$ is $\{0\}$.

Answer 2

The rows of $X^T$ span, so $X^Tv=0$ just when $v=0$, so $XX^Tv=0$ just when $v=0$, so $XX^T$ has kernel $0$, and is square, so is invertible.