1. ## Subspace

Suppose $\displaystyle U_1$ and $\displaystyle U_2$ be subspaces of a vector space V. Define the sum of $\displaystyle U_1+U_2$ as follows:
$\displaystyle U_1+U_2$ = {$\displaystyle u_1+u_2$: $\displaystyle u_1$ is an element of $\displaystyle U_1$, $\displaystyle u_2$ is an element of $\displaystyle U_2$}

1. Prove that $\displaystyle U_1+U_2$ is a subspace of V.

2. Let W be a subspace of V such that $\displaystyle U_1$ subset or equal to W and $\displaystyle U_2$ subset or equal to W. Prove that $\displaystyle U_1+U_2$ subset or equal to W

Thanks

2. Originally Posted by apple12
Suppose $\displaystyle U_1$ and $\displaystyle U_2$ be subspaces of a vector space V. Define the sum of $\displaystyle U_1+U_2$ as follows:
$\displaystyle U_1+U_2$ = {$\displaystyle u_1+u_2$: $\displaystyle u_1$ is an element of $\displaystyle U_1$, $\displaystyle u_2$ is an element of $\displaystyle U_2$}

1. Prove that $\displaystyle U_1+U_2$ is a subspace of V.
clearly $\displaystyle (U_1 + U_2) \subseteq V$. it remains to show

(1) $\displaystyle U_1 + U_2$ is closed under addition and (2) closed under scalar multiplication

do you know what that means? can you show it?

2. Let W be a subspace of V such that $\displaystyle U_1$ subset or equal to W and $\displaystyle U_2$ subset or equal to W. Prove that $\displaystyle U_1+U_2$ subset or equal to W

Thanks
You need to show that $\displaystyle x \in (U_1 + U_2) \implies x \in W$

can you show this?