Vector SpacesDate: 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/ |
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