How to show that an $n$-dimensional vector space over field $\mathbb{F}_p$ has $p^n$ elements. Suppose $V$ is an $n$-dimensional vector space over the finite field $\mathbb{F}_p$ for some prime $p$. How do I show that $V$ has $p^n$ elements? I was thinking that considering $n$ basis elements one can show that each element of the basis spans its own subspace which doesn't have any other basis elements, but how do I show that there are exactly $p^n$ elements overall?

Since $V$ is $n$-dimensional over $\mathbb{F}_p$, any given element $v\in V$ can be written as $$v=\lambda _1 v_1 + \cdots +\lambda _nv_n$$
for some $\lambda _1,\dots ,\lambda _n \in \mathbb{F}_p$, where $\{v_1 ,\dots ,v_n\}$ is a basis of $V$ over $\mathbb{F}_p$. For each $v_i$ you have $p$ possibilities for $\lambda _i$, it follows that, since $\{v_1 ,\dots ,v_n\}$ is a basis, you have $p^n$ such $v$'s.
