Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 5 down vote accepted

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.

share|cite|improve this answer

The following steps lead to a proof:

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

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

  3. Conclude your problem.

share|cite|improve this answer

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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.