TITLE: Span of union of subspaces
QUESTION [5 upvotes]: I was reading a proof in my linear algebra notes, and it says:
Let $U, W$ be subspaces of $\mathbb{R}^3$, $U = \text{span}\{(1,0,1), (0,1,1)\}$, $W = \text{span}\{(1,1,0), (0,1,1)\}$
Then it goes on to say $\text{span}(U \cup W) = \text{span} \{ (1,0,1), (0,1,1), (1,1,0) \}$
Why does the span of the union of $U$ and $W$ equal the span of the union of the sets that span $U$ and $W$?

REPLY [3 votes]: One direction is clear.  For the other direction, let $v\in$span$(U\cup W)$.  Then we may write $$v=a_1u_1+a_2u_2+\cdots+a_ku_k+b_1v_1+b_2v_2+\cdots+b_jv_j$$
where the $a_i, b_i\in\mathbb{R}$, $u_i\in U$, and $v_i\in W$.  But then we observe that 
$$a_1u_1+a_2u_2+\cdots+a_ku_k\in U$$
and 
$$b_1v_1+b_2v_2+\cdots+b_jv_j\in W$$
so indeed $v$ is the sum of a vector in $U$ and one in $W$.  But each vector in $U$ is the linear combination of  your three spanning vectors, as is each vector in $W$.  Hence $v$ is as well.