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### 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)

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

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

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/
```
Associated Topics:
High School Linear Algebra

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