Vector Spaces

Date: 06/11/99 at 00:04:19
From: Brian Reid
Subject: Vector spaces

Is V = {(x,y) in R^2 | y = 3x+1} a vector space if addition and 
multiplication by a scalar are defined by:

   (x,y) + (x',y') = (x+x',y+y'-1)
   k(x,y) = (kx,k(y-1)+1)

Part (b): give reasons for your answer in (a).

My son is taking math in summer session at the university and had 
trouble with this question on his last assignment. I tried to help him 
with it but couldn't. Can you show me how to do this one so I can help 
him out when he comes home next weekend?

Thank you very much,
Brian

Date: 06/11/99 at 05:40:22
From: Doctor Mitteldorf
Subject: Re: Vector spaces

Dear Brian,

I remember when I was a freshman (1966) and heard the term "vector 
space" for the first time, I couldn't help but carry with me all the 
ideas about vectors that I had acquired in physics problems, and that 
kept me confused for a few weeks. I broke out of that when I realized 
that the mathematician's idea of a vector space is exactly what it's 
defined to be: a set of objects in which you can guarantee that if you 
perform the operations of addition or scalar multiplication on them, 
you end up with another such object.

Actually, there's one more thing you need to assure: that scalar 
multiplication is distributive over addition.

The conventional wisdom is that a line or a plane passing through the 
origin is a natural vector space, but one that is skewed from the 
origin, as we have here, is not. For example, using the natural 
definition of addition, if you took two points that satisfied (y = 
3x+1) and added them up, you wouldn't get another point that satisfies 
(y = 3x+1). But the definitions of multiplication and addition 
provided here are intended to fudge around this problem, by relating 
back to the parallel line (y = 3x), adding, then shifting back over by 
1. Do they work as advertised? That's what the problem is intended to 
ask.

Why don't you try working it out yourself? Here are the three things 
you are to verify:

1) If (x1,y1) and (x2,y2) both satisfy (y = 3x+1) and they are added 
   according to the prescription given, then the result also satisfies 
   (y = 3x+1).

2) If (x1,y1) satisfies (y = 3x+1) and it is multiplied by a scalar
   according to the given definition of multiplication, then the
   result also satisfies (y = 3x+1).

3) Finally, verify that, with the given definitions for addition and
   scalar multiplication, k(A+B) = kA + kB, where k is a scalar and A
   and B are vectors.

Will you write and let me know what you find?

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/