Exercise 1.3.23

Question

Let $\mathrm{W}_{1}$ and $\mathrm{W}_{2}$ be subspaces of a vector space $\mathrm{V}$.
  \begin{enumerate}[label=(\alph*)]
\item Prove that $W_{1}+W_{2}$ is a subspace of $\mathrm{V}$ that contains both $W_{1}$ and $\mathrm{W}_{2}$.
\item Prove that any subspace of $\mathrm{V}$ that contains both $\mathrm{W}_{1}$ and $\mathrm{W}_{2}$ must also contain $\mathrm{W}_{1}+\mathrm{W}_{2}$.
\end{enumerate}

Jephian C.-H. Lin · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item \begin{proof} We have $(x_1+x_2)+(y_1+y_2)=(x_1+y_1)+(x_2+y_2)\in W_1+W_2$ and $c(x_1+x_2)=cx_1+cx_2\in W_1+W_2$ if $x_1,y_1\in W_1$ and $x_2,y_2\in W_2$. And we have $0=0+0\in W_1+W_2$. Finally $W_1=\{x+0:x\in W_1, 0\in W_2\}\subset W_1+W_2$ and it's similar for the case of $W_2$. \end{proof}
\item \begin{proof} If $U$ is a subspace contains both $W_1$ and $W_2$ then $x+y$ should be a vector in $U$ for all $x\in W_1$ and $y\in W_2$. \end{proof}
\end{enumerate}

Exercise 1.3.23

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Exercise 1.3.23

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