Consider the field $\mathbb{F}_p$ for $p$ prime. Find the number of subspaces of dimension 1 and 2 of the three dimenstional vector space $\mathbb{F}_p^3$. I know that $\mathbb{F}_p$ is generated by all elements relatively prime to $p$, but two subspaces are supposed to be equal if their bases can be expressed as linear combinations of each other. So aren't all 1 dimensional subspaces of $\mathbb{F}_p^3$ isomorphic to each other and in particular $\mathbb{F}_p$? Similarly for subspaces of dimension 2, you need two elements to generate a subset but all are linear combinations of one another so there can only be one.
This doesn't sound right, but I'm not sure what to do. I can't seem to find any information that would help figure this out.