Exercise 1.7.4

Question

Let $\mathbf{X}$ and $\mathbf{Y}$ be subspaces of a vector space $\mathbf{V}$. Using the previous exercise, show that $\mathbf{X} \cup \mathbf{Y}$ is a subspace if and only if $\mathbf{X} \subset \mathbf{Y}$ or $\mathbf{Y} \subset \mathbf{X}$.

Jing Huang · Accepted Answer

\begin{proof}
The sufficiency is obvious and easy to verify. For the necessity, suppose $\mathbf{X} 
subseteq \mathbf{Y}$ nor $\mathbf{Y} 
subseteq \mathbf{X}$ and $\mathbf{X} \cup \mathbf{Y}$ is a subspace of $\mathbf{V}$. Then there are vectors $\mathbf{x} \in \mathbf{X}$, $\mathbf{y} \in \mathbf{Y}$ and $\mathbf{x} 
otin \mathbf{Y}$, $\mathbf{y} 
otin \mathbf{X}$. According to Problem 7.3, $\mathbf{x} + \mathbf{y} 
otin \mathbf{X}$,  $\mathbf{x} + \mathbf{y} 
otin \mathbf{Y}$. So, $\mathbf{x} + \mathbf{y} 
otin \mathbf{X} \cup \mathbf{Y}$. i.e., 
$\mathbf{x} \in \mathbf{X} \cup \mathbf{Y}$, $\mathbf{y} \in \mathbf{X} \cup \mathbf{Y}$, but $\mathbf{x} + \mathbf{y} 
otin \mathbf{X} \cup \mathbf{Y}$, which contradicts $\mathbf{X} \cup \mathbf{Y}$ is a subspace. Thus, $\mathbf{X} \subset \mathbf{Y}$ or $\mathbf{Y} \subset \mathbf{X}$.
\end{proof}

Exercise 1.7.4

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Exercise 1.7.4

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