Linear Algebra

142 Chapter 2. Vector Spaces
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. (This says “almost all” 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 �v ∈ V and a ∈F, (i) 0 · �v = �0, and (ii) −1 · �v + �v = �0, and
(iii) a · �0 =�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
1 Show that the real numbers form a field.
2 Prove that these are fields:
(a) the rational numbers (b) the complex numbers.
3 Give an example that shows that the integer number system is not a field.
4 Consider the set {0, 1} subject to the operations given above. Show that it is a
field.
5 Come up with suitable operations to make the set {0, 1, 2} afield.