Exercise 1.3.17

Question

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

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
There is only one condition different from that of Theorem 1.3. If $W$ is a subspace, then $0\in W$ implies $W
eq \emptyset$. If $W$ is a subset satisfying the conditions of this question, then we can pick $x\in W$ since it's not empty and the other condition assures that $0x=0$ will be an element of $W$.
\end{proof}

Exercise 1.3.17

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Exercise 1.3.17

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