If $V/W\cong C$ is it true that $V=W\oplus C$ ?

Question

We know that for a vector space $V$ and its subspace $W$ if $V=W\oplus C$ then the quotient space $V/W$ is isomorphic to a subspace of $V$ (namely, $C$). Is the inverse true?

Answer 1

If you can find a short exact sequence for the subspace $C$ with maps $q$ and $r$ such that:

$ 0 \rightarrow W \overset{q}{\longrightarrow} V \overset{r}{\longrightarrow} C \rightarrow 0$

then one can find the additional arrows $t$ and $u$ (because every short exact sequence of vector spaces splits) such that:

$ 0 \rightarrow W {{q \atop \longrightarrow} \atop {\longleftarrow \atop t}} V {{r \atop \longrightarrow} \atop {\longleftarrow \atop u}} C \rightarrow 0. $

and that satisfies one of the following equivalent conditions:

1. left split: the map $t: V → W$ is such that $tq$ is the identity on $W$,
2. right split: the map $u: C → V$ is such that $ru$ is the identity on $C$,
3. direct sum: $V\cong W\oplus C$ with $q$ corresponding to the natural injection of $W$ and $r$ corresponding to the natural projection onto $C$.

Having said that it is important to remark as stated in the comments by Santiago Canez that if $V/W$ has the same dimension as $W$ they are isomorphic and not necessarily a direct sum of $V$.

A concrete example is $\mathbb{R}^{2}/span(1,0)$ this quotient is isomorphic to $span{(1,0)}$ ( because both are vector spaces of dimension 1). However,it is clear that $\mathbb{R}^{2}\neq span{(1,0)}+ span{(1,0)}$