# What does the sum of subsets of a vector space mean?

On page $57$ of Second edition of Hoffman Kunze, the authors write

Definition If $S_1, S_2, \dots, S_k$ are subsets of a vector space $V$, the set of all sums $$\alpha_1 + \alpha_2 + \dots+ \alpha_k$$ of vectors $\alpha_i$ in $S_i$ is called the sum of the subsets $S_1, S_2, \dots, S_k$ and is denoted by $$S_1+S_2+\dots+S_k$$ or by $$\sum_{i=1}^k S_k$$

But I have no idea what this means. What are the $\alpha$'s? Are they a vector from each subset? If yes then which one? If not then what are they? I can't understand what is meant by '$\alpha_i$ in $S_i$'.

I know of Minkowski Sum of 2 sets. Is this something similar?

• @M.Vinay I know that $\alpha_i$ are vectors. I wanted to know from where, that is, whether they are any vector from $S_i$ or a particular vector, so on and so forth. – Aritra Das May 31 '16 at 3:55
• @M.Vinay I'm sorry but you sound a bit too condescending for me. Anyway, thanks for your inputs. – Aritra Das May 31 '16 at 4:14
• I am sorry, I didn't mean to be. I will delete my comments. – M. Vinay May 31 '16 at 4:15

Yes, for each $i$, the vector $\alpha_i$ comes from $S_i$. Yes, this is the same as the Minkowski sum of $S_1,\ldots,S_k$.
Yes, this is just the generalization of Minkowski sum to $k$ sets. For each $i$ from $1$ to $k$, $\alpha_i\in S_i$. That is, $$S_1+S_2+\dots+S_k=\{\alpha_1+\alpha_2+\dots+\alpha_k:\alpha_1\in S_1,\alpha_2\in S_2,\dots,\alpha_k\in S_k\}.$$