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In A Generic Approach to Invariant Subspace Attacks the authors defined an invariant subspace as: $$F(u + A) = v + A$$

Where $F:\mathbb{F}_2^n \to \mathbb{F}_2^n$,$u, v \in \mathbb{F}_2^n$, and $A$ is a linear subspace.

Then they claimed that if $M\circ F = F\circ M$ for a linear subspace $M$ then $A:=ker\left(M+Id\right)$ is a linear subspace(Lemma 2).

I couldn't figure out why this is true for any bijective $F$ function. Any idea how to prove it?

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    $\begingroup$ This question would probably be a better fit for Mathematics, as it seems to be pure linear algebra. $\endgroup$ – Ilmari Karonen Jan 26 at 19:08

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