Sum and union of subspaces

Mar 17, 2025 3:47 PM

Give an explicit example (with proof) showing that the union of two subspaces (of a given vector space) is not necessarily a subspace.
Suppose $U_{1}$ and $U_{2}$ are subspaces of a vector space $V$ . Recall that their sum is defined to be the set $U_{1} + U_{2} = {u_{1} + u_{2} ∣ u_{1} \in U_{1}, u_{2} \in U_{2}}$ . Prove $U_{1} + U_{2}$ is a subspace of $V$ containing $U_{1}$ and $U_{2}$ .

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\begin{enumerate}[label=\alph*)]
	\item Give an explicit example (with proof) showing that the union of two subspaces (of a given vector space) is not necessarily a subspace.
	\item Suppose $U_1$ and $U_2$ are subspaces of a vector space $V$. Recall that their {\bfseries sum} is defined to be the set $U_1+U_2 =\left\{u_1+u_2\,\mid \, u_1\in U_1, u_2\in U_2\right\}$. Prove $U_1+U_2$ is a subspace of $V$ containing $U_1$ and $U_2$.
\end{enumerate}