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I read the answer to this question, but the answer wasn't very clear to me. My question is similar, so I state my question. Could someone clarify the answer to that question, and/or to my question? (If I understand it right, my question is a somewhat more specific case of the other one, right?)

Given an $n$-dimensional vectorspace over a finite field ${\mathbb{F}_q}$, how many solutions are there to a system of $k$ linear equations of full rank (rank $k$) in $n$ variables? (assume $k~<~n$)

(Sorry if the answer to the other question seems clear and the question too simple, but I'm still confused.)

nara
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1 Answers1

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The set of solutions is given by $x_0+W$ where $W$ is a subspace of the given vector space of dimension $m:=n-k$. Let $\{w_1,w_2,\ldots,w_m\}$ be a basis of $W$. Then the map $f:\mathbb{F}_q^m\to x_0+W$, $f((a_1,a_2,\ldots,a_m)):=x_0+\sum_{i=1}^m a_i w_i$, is a bijection. Note finally that $\mathbb{F}_q^m$ has $q^m$ elements.

Jens Schwaiger
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