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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.

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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 $.

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