Linear Algebra, Topic: Fields

Linear combinations involving only fractions or only integers are much
easier for computations than combinations involving real numbers,
because computing with irrational numbers is awkward.
Could other number systems, like the rationals or the integers, work in the
place of ℝ in the definition of a vector space?

Yes and no.
If we take "work" to mean that the results of this chapter
remain true
then an analysis of which properties of the reals we have
used in this chapter gives the following list of conditions an algebraic system
needs in order to "work" in the place of ℝ.

Definition 1.1

A field is a set ℱ with two operations"+" and "⋅" such that

• for any a , b ∈ ℱ the result of a + b is in ℱ and

a + b = b + a

• if c ∈ ℱ then a + ( b + c ) = ( a + b ) + c
• for any a , b ∈ ℱ the result of a ⋅ b is in ℱ and

a ⋅ b = b ⋅ a

• if c ∈ ℱ then a ⋅ ( b ⋅ c ) = ( a ⋅ b ) ⋅ c
• if a , b , c ∈ ℱ then a ⋅ ( b + c ) = a ⋅ b + a ⋅ c
• there is an element {}0 ∈ ℱ such that

if a ∈ ℱ then a + 0 = a

• for each a ∈ ℱ there is an element − a ∈ ℱ
  such that ( − a ) + a = 0
• there is an element {}1 ∈ ℱ such that

if a ∈ ℱ then a ⋅ 1 = a

• for each element a ≠ 0 of ℱ
  there is an element {a}^{− 1} ∈ ℱ
  such that {a}^{− 1} ⋅ a = 1.

The number system consisting of the set of real numbers along with the usual
addition and multiplication operation is a field, naturally.
Another field is the set of rational numbers with its usual addition
and multiplication operations.
An example of an algebraic structure that is not a field is
the integer number system — it fails the final condition.

Some examples are surprising.
The set \{ 0 , 1 \} under these operations:

\begin{matrix}+ & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 1 & 0\end{matrix} \ \begin{matrix}⋅ & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1\end{matrix}

is a field (see Problem 4).

We could develop Linear Algebra as the theory of
vector spaces with scalars from an arbitrary field,
instead of sticking to taking the scalars only from ℝ.
In that case,
almost all of the statements in this book would carry over by replacing
"ℝ" with "ℱ", and thus by
taking coefficients, vector entries,
and matrix entries to be elements of ℱ
("almost" because statements involving distances or angles
are exceptions).
Here are some examples; each applies to a vector space V
over a field ℱ.

• For any \overset{→}{v} ∈ V and a ∈ ℱ,

{}0 ⋅ \overset{→}{v} = \overset{→}{0}, and

• − 1 ⋅ \overset{→}{v} + \overset{→}{v} = \overset{→}{0}, and
• a ⋅ \overset{→}{0} = \overset{→}{0}.
• The span (the set of linear combinations) of a subset of V

is a subspace of V.

• Any subset of a linearly independent set is also
  linearly independent.
• In a finite-dimensional vector space,

any two bases have the same number of elements.

(Even statements that don't explicitly mention ℱ use
field properties in their proof.)

We won't develop vector spaces in this more general setting because
the additional abstraction can be a distraction.
The ideas we want to bring out already appear when we stick to the reals.

The only exception is in Chapter Five.
In that chapter we must factor polynomials,
so we will switch to considering vector spaces over the
field of complex numbers.
We will discuss this more, including a brief review of complex arithmetic,
when we get there.

Exercises

Problem 1

Show that the real numbers form a field.

Problem 2

Prove that these are fields.

• The rational numbers ℚ
• The complex numbers ℂ

Problem 3

Give an example that shows that the integer number system is not a field.

Problem 4

Consider the set \{ 0 , 1 \} subject to the operations given above. Show that it is a field.

Problem 5

Give suitable operations to make the set \{ 0 , 1 , 2 \} a field.

Solutions

Where this page came from

This page was imported from Wikibooks. From “Linear Algebra” on Wikibooks, by its contributors, under CC BY-SA 4.0. Changed here: set as a page; navigation and edit links left out; each image under its own licence, credited in its caption.

Nobody has written it yet — it is the source material at a new address, which is why search engines are asked to skip it and why no one earns from it. It is up for grabs: take it on, and it is yours to rewrite and to earn from.