Exercise 1.3.19

Question

Let $W_{1}$ and $W_{2}$ be subspaces of a vector space $V$. Prove that $W_{1} \cup W_{2}$ is a subspace of $V$ if and only if $W_{1} \subseteq W_{2}$ or $W_{2} \subseteq W_{1}$.

Jephian C.-H. Lin · Accepted Answer

\begin{proof}
It's easy to see that is sufficient since if we have $W_1\subset W_2$ or $W_2\subset W_1$ then the union of $W_1$ and $W_2$ will be $W_1$ or $W_2$, a space of course. To prove that the condition is necessary, we may assume that neither $W_1\subset W_2$ nor $W_2\subset W_1$ holds; then we can find some $x\in W_1\backslash W_2$ and $y\in W_2\backslash W_1$. Thus, by the condition of subspace, we have $x+y$ is a vector in $W_1$ or in $W_2$, say $W_1$. But this will make $y=(x+y)-x$ should be in $W_1$. It will be contradictory to the original hypothesis that $y\in W_2\backslash W_1$.
\end{proof}

Exercise 1.3.19

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Exercise 1.3.19

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