Let $V$ be a three dimensional vector space over $\mathbb Z_3$. Then the problem is to count the distinct one dimensional subspaces. $\mathbb R^3 $ has infinitely many one dimensional subspaces. If $v_1, v_2, v_3$ are a basis, any nontrivial linear combination of these three vectors leads to a one dimensional subspace. How to isolate the distinct ones?
[Math] Number of one dimensional subspaces
linear algebra
Best Answer
Hint: How many non-zero vectors are there? When do two non-zero vectors determine the same subspace?
Remark: In general, take a finite field $F$ with $k$ elements, and look at the one-dimensional subspaces of an $n$-dimensional space over $F$. Then there are $k^n-1$ non-zero vectors. For any vector $v$, there are $k-1$ vectors that determine the same subspace as $v$ (multiply $v$ by any non-zero element). So the number of one-dimensional subspaces is $\frac{k^n-1}{k-1}$. We can extend this analysis to count the $d$-dimensional subspaces.