# Understanding a question about vector space and subspace

Let $$V$$ be a vector over a field $$F$$. On the set $$S$$ of subspaces of $$V$$, define an addition by setting

$$U_1 + U_2 = \{u_1+u_2: u_1 \in U_1, u_2 \in U_2 \}$$.

And I'm supposed to prove the vector space axioms only for addition, but before even trying to prove them I can't understand how the addition is defined. I think I am having trouble with the English not the math. If someone wouldn't mind interpreting and clarifying it for me it will be greatly appreciated.

I'm assuming the sets $$U_1, U_2$$ are in $$S$$? The set is defined as all vectors that are a sum of a vector in $$U_1$$ and $$U_2$$. For example, let $$V = \mathbb{R}^2, U_1 = \mathrm{span} \left( \left \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \right \} \right), U_1 = \mathrm{span} \left( \left \{ \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right \} \right)$$
You can see that $$U_1 + U_2 = \mathbb{R}^2$$ by noting that $$\begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} a \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ b \end{bmatrix}$$ and $$\begin{bmatrix} a \\ 0 \end{bmatrix} \in U_1, \begin{bmatrix} 0 \\ b \end{bmatrix} \in U_2$$.