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?

  • $\begingroup$ 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? $\endgroup$ – Eric Wofsey Mar 15 at 7:02

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