Let $V=U \bigoplus W$, $V \approx U \times W$. Note that $U,W$, are finite dimensional subspaces of the vector space V, and also that $U \bigoplus W$ means $V=U+W$ and $U \cap W = \{0\}$
I'm really not sure how to go about this, because it doesn't seem to be true to me. But after some research, it does seem to be true. Thanks in advance.
You can construct the isomorphism explicitly. If $V = U \oplus W$ then any $v \in V$ can be written uniquely as $u+w$ for $u \in U$ and $w \in W$. Define $f : V \to U \times W$ by $u+w \mapsto (u,w)$. It is easy to check that this is a well-defined linear isomorphism.
The following steps lead to a proof:
Any two finite dimensional vector spaces of the same dimension are isomorphic.
$\dim (U \oplus W) =\dim (U \times W)$.
Conclude your problem.
By definition
$$ U \oplus V = \left\{ u + v \ \vert \ u\in U, v \in V \right\} \ , $$
plus the fact that $U\cap V = \left\{ 0\right\}$.
Also by definition
$$ U\times V = \left\{ (u,v) \ \vert \ u\in U, v \in V \right\} \ . $$
Can we conclude anything from that in order to define isomorphisms
$$ f: U\oplus V \longrightarrow U\times V \qquad \text{and} \qquad g: U\times V \longrightarrow U\oplus V \quad \text{?} $$
