If $F$ is a finite field and $V$ is a $n$-dimensional vector space over $F$ then we have to find the cardinality of the set $$S=\{f\in End(V) : fv=v \text{ for some non-zero }v\}$$
I have no clue where to start from. Any hint would be appreciated
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2First observe that $f\in End (V)$ fixes a non zero vector iff $f-Id$ is singular. So it suffices to count the number of singular endomorphisms. – user6 May 07 '20 at 18:24
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1thanks, got it! – Baidehi May 07 '20 at 18:35
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I have missed you thought twice. This is the third time. This is equivalent to finding matrix $(A-I)x =0$ for some $x$ in $F^n$. Consequently, the cardinality is the subset $H$ consisted by all matrices $A$ which satisfies that $rank(A-I) ＜ n$.
Upgradation: The key of question is to find how many invertible matrices. This has been solved. Look over here
This address is being provided by the order of elements in $GL_{n}(F)$ over a field.
Mod.esty
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But I'm not looking to find the cardinality of End(V). I'm looking to find the cardinality of the subset of End(V) that fixes at least one non-zero vector. – Baidehi May 07 '20 at 18:13
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@User2592 I am not sure what do you mean "fix at least...". You mean if there is a non-zero vector $a$, you want the cardinality of a subset $A$ which has $a$? – Mod.esty May 07 '20 at 18:24
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@User2592 or you mean a subset $A$(not a subspace) need to contain at least one non-zero vector? – Mod.esty May 07 '20 at 18:26