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Let $\Bbb{F}_2^n$ be $n$-dimensional vector space over $\Bbb{F}_2$, the two element field.

Approximately how many distinct subspaces $U\le \Bbb{F}_2^n$ of codimension 1 (i.e., dimension $n-1$) are there for large $n$? More generally, for fixed $k$, how many distinct subspaces $U\le \Bbb{F}_2^n$ of codimension $k$ (i.e., dimension $n-k$) are there for large $n$? Here, subspaces $U,U'$ are considered distinct if they have different sets of elements, (i.e., we don't care about their bases or anything).

Zach Hunter
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    Do you know that a subspace of $\mathbb R^n$ of codimension $k$ is the null space of a matrix with $k$ linearly independent rows? The proof of that works over arbitrary fields with no changes -- and of course when $k=1$, "linearly independent" is just "non-zero". So how many nonzero $1\times n$ matrices over $\mathbb F_2$ are there? Do any of them have the same null space? – Troposphere Jul 29 '21 at 13:41
  • This is a duplicate. Many voters want to close this for other reasons, so I don't want to overrule them with my dupehammer. – Jyrki Lahtonen Jul 31 '21 at 08:10

1 Answers1

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The set of all hyperplanes is in bijection with the set of lines of the dual space, which is isomorphic to $\mathbb{F}_2^n$. So you have exactly $$2^n-1$$

hyperplanes in $\mathbb{F}_2^n$.

TheSilverDoe
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