1
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ @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. $\endgroup$ – Aritra Das May 31 '16 at 3:55
  • $\begingroup$ @M.Vinay I'm sorry but you sound a bit too condescending for me. Anyway, thanks for your inputs. $\endgroup$ – Aritra Das May 31 '16 at 4:14
  • $\begingroup$ I am sorry, I didn't mean to be. I will delete my comments. $\endgroup$ – M. Vinay May 31 '16 at 4:15
3
$\begingroup$

Yes, for each $i$, the vector $\alpha_i$ comes from $S_i$. Yes, this is the same as the Minkowski sum of $S_1,\ldots,S_k$.

$\endgroup$
3
$\begingroup$

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, $$S_1+S_2+\dots+S_k=\{\alpha_1+\alpha_2+\dots+\alpha_k:\alpha_1\in S_1,\alpha_2\in S_2,\dots,\alpha_k\in S_k\}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.