# Understanding Vector Spaces

Discussion in 'General Math' started by Ken, Aug 15, 2007.

1. ### KenGuest

I answered the following two questions as true...
a) in any vector space ax=bx implies that a = b. Where a,b are real
numbers and x is a vector.
b) In any vector space ax = ay implies that x = y. Where a is a real
number and x and y are vectors.
How can these be false?
and there is one more I don't follow...
c) If f and g are polynomials of degree n, then f + g is a polynomial
of degree n. (I also said this was true but it seems the answer is
false).

If you wish to reference the conditions of vector spaces... [1-8] or
simply state that a is in violation of any of these conditions it is
here for reference: http://en.wikipedia.org/wiki/Vector_spaces

Ken, Aug 15, 2007

2. ### Brian VanPeltGuest

For a) let x be the zero vector. For b) let a be the zero scalar.

For part c) suppose f(x) = x, and g(x) = -x, what is the degree of
f+g?

Brian

Brian VanPelt, Aug 16, 2007

3. ### KenGuest

Thank you for the help... I will from now on all ways consider the
zero case.
The degree is -1.

Ken, Aug 30, 2007
4. ### Brian VanPeltGuest

Wow, it's been a long time since I have written this message and I

This brings up an interesting point though - the degree of the zero
polynomial. Did you know that some people don't bother to define the
degree of the zero polynomial? But yet, some people give it degree
-1, as you do? Could it be -2, -pi, or otherwise? Could the degree
of the zero polynomial be i (ooh, i has magnitude equal to 1)? Every
stance has its reasons, but I have never seen one stance that
logically crushes all others, and it's quite possible that there is no
logically correct stance here.

Thanks,

Brian

Brian VanPelt, Aug 31, 2007