Page 107 - 35Linear Algebra

5.2 Other Fields 107

Example 66 P := a a, b ≥ 0 is not a vector space because the set fails (·i)
b

1 1 −2
since ∈ P but −2 = / ∈ P.
1 1 −2
S
Sets of functions other than those of the form R should be carefully
checked for compliance with the definition of a vector space.

Example 67 The set of all functions which are nowhere zero

{f : R → R | f(x) 6= 0 for any x ∈ R} ,

does not form a vector space because it does not satisfy (+i). The functions f(x) =
2
2
x +1 and g(x) = −5 are in the set, but their sum (f +g)(x) = x −4 = (x+2)(x−2)
is not since (f + g)(2) = 0.

5.2 Other Fields

Above, we defined vector spaces over the real numbers. One can actually
define vector spaces over any field. This is referred to as choosing a different
base field. A field is a collection of “numbers” satisfying properties which are
listed in appendix B. An example of a field is the complex numbers,

2
C = x + iy | i = −1, x, y ∈ R .

Example 68 In quantum physics, vector spaces over C describe all possible states a
physical system can have. For example,

λ
V = | λ, µ ∈ C
µ

1 0
is the set of possible states for an electron’s spin. The vectors and describe,
0 1
respectively, an electron with spin “up” and “down” along a given direction. Other

i
vectors, like are permissible, since the base field is the complex numbers. Such
−i
states represent a mixture of spin up and spin down for the given direction (a rather
counterintuitive yet experimentally verifiable concept), but a given spin in some other
direction.

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