# Fix $v$, the probability that $v$ is orthogonal to any proper subspace of a random matrix $A$ with standard normal entries is $0$

Suppose $$v \in \mathbb R^n$$ is some fixed vector and $$A \in M_n(\mathbb R)$$ is a random matrix that each entry is generated by a standard normal distribution. I read a statement claiming: the probability that $$v$$ is orthogonal to any proper subspace of column space of $$A$$ is $$0$$. How do we prove this? Is there a good reference that I can read related stuff?

• This is trivially false: $v$ is orthogonal to $\{0\}$, which is a proper subspace of the column space of $A$ unless $A=0$. Perhaps you meant to say something different? – Eric Wofsey Mar 15 at 7:02