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.

Thank you in advance.

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 $$\begin{bmatrix} a \\ 0 \end{bmatrix} \in U_1, \begin{bmatrix} 0 \\ b \end{bmatrix} \in U_2$$.