I’ve been going over some old textbooks and two of them, a matrix text and a linear algebra text prove many theorems using summations.
As an example, two of the rules they cite for manipulating summations are:
A factor independent of the index of summation may be introduced under the summation sign.
The order of summation in a finite double sum can be commuted.
Where did these rules come from, and what kind of text would I use to familiarize myself with these manipulations? I couldn’t find a website that covers this stuff, but I may not be using the right keywords.
That would be an appropriate text if the OP were asking about infinite series, but I believe Ring just wants to understand how plain finite sums, when expressed using the capital sigma symbol, are manipulated.
Ring, the rules you mention follow directly from the basic properties of addition: commutativity and distributively. The best way to see the rules for manipulating summations expressed with the sigma sign is just to write out what they mean using plain ‘+’ signs. Someone here who has more skill using the symbol font than I have may be able to render this on this message board.
I copied “properties of finite sums” into Google and got a couple of hits that looked somewhat promising. Unfortunately they were both in Postscript and I don’t know what that is and neither does Netscape…so I’m doomed.
These kinds of theorems are generally not that difficult to prove. Try writing the statement of the theorem without using sigma notation, and there’s a good chance it’ll become clear what the theorem is saying.
In particular, the two theorems you mentioned are really just statements of distributivity and commutativity (as Tyrrell McAllister mentioned). This is fairly easy to see if you write the statement without the sigma notation.