Exercise 1.3.18

Question

Prove that a subset $\mathrm{W}$ of a vector space $\mathrm{V}$ is a subspace of $\mathrm{V}$ if and only if $0 \in \mathrm{W}$ and $a x+y \in \mathrm{W}$ whenever $a \in F$ and $x, y \in W$.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
We compare the conditions of this exercise with the conditions of Theorem 1.3. First let $W$ be a subspace. We have $cx$ will be contained in $W$ and so is $cx+y$ if $x$ and $y$ are elements of $W$. Second let $W$ is a subset satisfying the conditions of this question. Then by picking $a=1$ or $y=0$ we get the conditions in Theorem 1.3.
\end{proof}

Exercise 1.3.18

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Exercise 1.3.18

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