Is quotient space a subspace of the underlying vector space

Question

If $W$ is a subspace of the finite dimensional vector space $V$, is the quotient space $V/W$ a subspace of $V$ ? How do I prove or disprove this statement ?

Answer 1

Since $V$ is finite dimensional, so are $W$ and $V/W$. Thus, we have $V \cong k^n$ and $V/W \cong k^m$ with $m \leq n$, where $n=\text{dim}(V)$, $m=n-\text{dim}(W)$ and $k$ denotes the underlying field. It follows that $$ V/W \cong k^m \hookrightarrow k^n \cong V.$$ Hence, $V/W$ may be regarded as a subspace of $V$.

However, in general, $V/W$ is not $\textbf{equal}$ to a subspace of $V$ since it is not even a subset of $V$.