Exercise I.C.9

Question

Let $V$ be a vector space, and let $U,W$ be vector subspaces of $V$. Show that
$$U,W 	ext{ are complementary } \iff 	ext{ each $v\in V$ may be uniquely represented as $v=u+w$ for $u\in U$, $w \in W$}$$

Toğrul Topal · Accepted Answer

\begin{itemize}
    \item[$\implies)$] Suppose that $U$ and $W$ are complementary. Then we can represent each $v \in V$ as $v = u+w$ for $u\in U, w \in W$ by definition. Now suppose that there is another $u'\in U$ and $w' \in W$ such that $v = u' + v'$. We then have $u' + w' = u + w \implies u' - u = w - w'$, or in other words, $w-w' \in U \cap W$ and $u-u' \in U \cap W$ - a contradiction to the fact that $U \cap W = \{0\}$.
    \item[$\impliedby)$] Again, $U+W = V$ follows by definition. Now suppose that $v \in U \cap W$. Then we have two distinct representations: $v = v + 0$ for $v\in U, 0 \in W$ and $v = 0 + v$ for $v \in W$ and $0 \in U$.
    \end{itemize}

Exercise I.C.9

Answers

Comments

Exercise I.C.9

Answers

Comments

Add answer