Suppose $V$ is an $n$-dimensional space on the field $\mathbb F_p$ and $W$ is an $m$-dimensional subspace of $V$

Question

Suppose $V$ is an $n$-dimensional space on the field $\mathbb F_p$ and $W$ is an $m$-dimensional subspace of $V$ where $m <n$.

Show that $p^{n − m}$ sets can be found, each of which is $ a_j + W = \{a_j + w: w ∈ W\} $, so that

$$ V = \bigcup_{i=1}^{p^{n-m}}a_i + W $$

Answer 1

Since $\dim(V/W)=\dim(V)-\dim(W)$, we have $\mid V/W | = p^{n-m}$. Since the elements of $V/W$ are equivalence classes, their union is $V$.