# Prove that $V=U \bigoplus W \approx U \times W$

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.

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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.

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The following steps lead to a proof:

1. Any two finite dimensional vector spaces of the same dimension are isomorphic.

2. $\dim (U \oplus W) =\dim (U \times W)$.

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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

$$f: U\oplus V \longrightarrow U\times V \qquad \text{and} \qquad g: U\times V \longrightarrow U\oplus V \quad \text{?}$$

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