Exercise 1.C.10

Question

Suppose $U_1$ and $U_2$ are subspaces of $V$. Prove that the intersection $U_1 \cap U_2$ is a subspace of $V$.

Célio Passos · Accepted Answer

\begin{proof}
Clearly $0 \in U_1 \cap U_2$.\par

Suppose $u, w \in U_1 \cap U_2$. It follows that $u, w \in U_1$. Because $U_1$ is closed under addition, $u + w \in U_1$. Likewise, $u + w \in U_2$. Hence $u + w \in U_1 \cap U_2$ and $U_1 \cap U_2$ is closed under addtion.\par

Similarly, $U_1 \cap U_2$ is closed under scalar multiplication, therefore it is a subspace of $V$.
\end{proof}

Exercise 1.C.10

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Exercise 1.C.10

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