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When solving a linear system for x and y, am i in a group? ring? field?

by ksmith630
Tags: field, linear, ring, solving
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May4-12, 06:21 PM
P: 606
Quote Quote by ksmith630 View Post

If you don't quote it is impossible to know whom you're addressing.

May4-12, 06:28 PM
P: 606
Quote Quote by micromass View Post
I don't see anybody claiming that your coefficients and your x,y,z need to be in the same space. I think it's an unnecessary assumption that they do.

Vector spaces are precisely there for being able to solve linear systems of equations. I can solve any linear system of equation while working in a vector space. Furthermore, even if the coefficients and the x,y,z are in the same space, then I'm still working in a vector space, a one-dimensional one.

From the example the OP presented I think it was clear that he/she was struggling with the idea that of what alg. structure her/his equations'

coefficients were to be taken from, and thus what operations and elements could be used. This isn't answered by working with vector spaces

as one can't multiply vectors in general vector spaces.

Vectors spaces provide, among other things, an adequate alg. frame in which we can work with linear equations, but the basic operations with vectors

usually have a field (or division ring) as base alg. structure from where the operations are taken. This is what I meant.

May4-12, 08:15 PM
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Quote Quote by HallsofIvy View Post
One way of solving that particular system would be to add the two equations to get
(x+ y)+ (x- y)= 2x= 5+ 1= 6 and then divide by 2. In order to add the two equations, you have to be working in at least a group but in order to divide by 2 you must we working in a ring or vector space.
It finally occured to me that while we can multiply in a field, we cannot necessarily divide. I should have said "field or vector space" rather than "ring or vector space".
May4-12, 10:04 PM
P: 10
Ohh OK i see now. I'll choose the field then- but how do i write the notation? I know if i was working in a group i'd write:

"To solve, i will be in the group <Z,+> where + is the ordinary binary operation in Z."

So for this system in the field, would i write:

"To solve, i will be in the field <Q,+,x> where + and x are the ordinary binary operations in Q." ?
May7-12, 02:27 PM
P: 10
does this sound right guys?
May7-12, 03:17 PM
P: 10
or what about if i worked in the ring of rationals?

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