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Suppose I know that the nullspace of matrix $P$ is $\mathbf{c}$. Now suppose I multiple $P$ by some matrix $C$, ie. $CP$. Is the nullspace of $CP$ also just $\mathbf{c}$? Or can there be other vectors in the nullspace not equal to $\mathbf{c}$?

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    $\begingroup$ Multiplying by the zero matrix should give you a clue. If zero is not allowed, you can still multiply by matrices which evaluate to zero when multiplied with most vectors. $\endgroup$ Mar 6, 2021 at 5:33
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    $\begingroup$ Ah that makes sense thanks! $\endgroup$
    – jlcv
    Mar 6, 2021 at 5:56
  • $\begingroup$ you are welcome! $\endgroup$ Mar 6, 2021 at 8:53

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$Px=0\Rightarrow CPx=0$ for any $P,~C$. So null space of $P$ is a subset of the null space of $CP$. They are not always equal, for example if $P$ is non-singular and $C$ is singular then since $CP$ is singular, the null space of $CP$ is strictly bigger.

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