I saw this property of gaussian vector : If $X$ is a random vector of normal distribution $\mathcal{N}(\mu, \Sigma)$, then $X$ takes values a.s. in $\{\mu + x \mid x \in \mathrm{Im}\Sigma^{1/2} \}$.
To prove this, we can use the fact that $X \sim \mu + \Sigma^{1/2}Z$ where $Z \sim \mathcal{N}(0, I_{n})$.
I think this propery suggest (implies) that if $A, B \in M_{n}(\mathbb{R})$ such that $AA^{T} = BB^{T}$, then $\mathrm{Im} A = \mathrm{Im} B$.
I'm not sure if this is correct and how to prove this directly from linear algebra?
