Exercise I.A.2

Question

If $U,W$ are subspaces of the vector space $V$, show that the sum of $U$ and $W$
$$U+W = \{u+w| u\in U, w\in W\}$$
is also a subspace of $V$.

Toğrul Topal · Accepted Answer

Again, we use Proposition 1. Let $c\in\mathbf{K}$ be arbitrary, and let $x,y\in U+ W$ be arbitrary as well. By definition, $x=u+w$ and $y = u' + w'$ for some $u,u' \in U$ and $w,w' \in W$. We then have $cx+y = c(u+w) + (u'+w') = cu + cw + u' + w' = (cu+u') + (cw+w')$, and since $cu+u'\in U$ and $cw+w'\in W$ we have $cx+y \in U+W$.

Exercise I.A.2

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Exercise I.A.2

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