Some Linear Algebra

Question

A vector space $V$ is said to be the direct sum of $U$ and $W$ if $U$ and $W$ are subspaces of $V$ such that $U\cap W =\{0\}$ and $U+W=V$. To denote that $V$ is the direct sum of $U$ and $W$, it is written as $V=U\oplus W$.

(a) Let $U$ and $W$ be $F$-vector spaces, where $F$ is any field. Let $U\times W=\{(u,w):u\in U, w\in W\}$ and define addition and scalar multiplication on $U\times W$ componentwise. Show that $U\times W$ is an $F$-vector space.

(b) Let $U$ and $W$ be subspaces of an $F$-vector space $V$, where $F$ is any field and suppose $V=U\oplus W$. Prove or disprove: $V\cong U\times W$.

There were other parts of the question that were more straightforward, but I wasn't sure how to start these ones. For (a), do I just have to show $U\times W$ satisfies the vector space axioms?

In (b) should I assume that a bijective homomorphism exists between the $U\times W$ and $V$? Where does the direct sum property come in for part (b)?

Answer 1

(a) Indeed you have to use the vector space axioms.

(b) If the sum is direct then:

• The map $\varphi : (u,w) \mapsto u + w$ is surjective as $U +W = V$ by hypothesis.
• And injective as $u + w = 0$ is equivalent to $u=-w$ which implies $u=w=0$ as $U \cap W = \{0\}$.

Finally $\varphi$ is an isomophism between $(U,W)$ and $V$, proving that $V\cong U\times W$.

Answer 2

For (a), yes, you should show that $U\times W$ satisfies the vector space axioms.

For (b), you should define an isomorphism from $U\oplus W$ onto $U\times W$ (or from $U\times W$ onto $U\oplus W$) or to prove that such an isomorphism doesn't exist.