(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that the number of distinct solutions of a system of linear equations (in any number of equations, and unknowns) over the field Zp is either 0, or a power of p.

3. The attempt at a solution

First off, I was wondering whether there is any difference between "cardinality" of a vector space and "dimension". Aren't both just the size of the basis? (the cardinality of V = dim(V)??)

This is just because my prof keeps switching between both and confusing the rest of us.

for the question, suppose the system is m equations in n unknowns.

The case of 0 is trivial, so if we take the subset W of Z_{p}^{n}of solutions over Z_{p}, W is a vector space, and dim(W) = n.

I'm not sure how to put this technically, but for each unknown more than the number of equations (say we parametrize them) we have p possible choices, and thus p^{t}solutions.

There's something missing, but am I on the right track?

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# Homework Help: Cardinality vs. Dimension, Solution of homogeneous equations

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