Let $A$ be similar to $B$ (so $A=PBP^{-1}$). If we have a vector $v$ that's in the column space of $A$, then how do we show that $P^{-1}v$ is in the column space of $B$? I'm trying to figure out how the column spaces between $A$ and $B$ relate to each other but I seem to be stuck.
2026-03-25 12:55:37.1774443337
Similar matrices and their column space
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$v$ being in the column space of $A$ means that there is some $u$ such that $Au = v$.
This means we have $PBP^{-1}u = v$.
The way to proceed is multiply both sides of the equation by $P^{-1}$. Can you see how to get to the answer from there?