Linear Algebra

122 Chapter 2. Vector Spaces
2.11 Corollary Any spanning set can be shrunk to a basis.
Proof. Call the spanning set S. If S is empty then it is already a basis. If
S = {�0} then it can be shrunk to the empty basis without changing the span.
Otherwise, S contains a vector �s1 with �s1 �= �0 and we can form a basis
B1 = 〈�s1〉. If[B1] =[S] then we are done.
If not then there is a �s2 ∈ [S] such that �s2 �∈ [B1]. Let B2 = 〈�s1, �s2〉; if
[B2] =[S] then we are done.
We can repeat this process until the spans are equal, which must happen in
at most finitely many steps. QED
2.12 Corollary In an n-dimensional space, a set of n vectors is linearly independent
if and only if it spans the space.
Proof. First we will show that a subset with n vectors is linearly independent
if and only if it is a basis. ‘If’ is trivially true — bases are linearly independent.
‘Only if’ holds because a linearly independent set can be expanded to a basis,
but a basis has n elements, so that this expansion is actually the set we began
with.
To finish, we will show that any subset with n vectors spans the space if and
only if it is a basis. Again, ‘if’ is trivial. ‘Only if’ holds because any spanning
set can be shrunk to a basis, but a basis has n elements and so this shrunken
set is just the one we started with. QED
The main result of this subsection, that all of the bases in a finite-dimensional
vector space have the same number of elements, is the single most important
result in this book because, as Example 2.9 shows, it describes what vector
spaces and subspaces there can be. We will see more in the next chapter.
2.13 Remark The case of infinite-dimensional vector spaces is somewhat controversial.
The statement ‘any infinite-dimensional vector space has a basis’
is known to be equivalent to a statement called the Axiom of Choice (see
[Blass 1984]). Mathematicians differ philosophically on whether to accept or
reject this statement as an axiom on which to base mathematics. Consequently
the question about infinite-dimensional vector spaces is still somewhat up in the
air. (A discussion of the Axiom of Choice can be found in the Frequently Asked
Questions list for the Usenet group sci.math. Another accessible reference is
[Rucker].)
Exercises
Assume that all spaces are finite-dimensional unless otherwise stated.
� 2.14 Find a basis for, and the dimension of, P2.
2.15 Find a basis for, and the dimension of, the solution set of this system.
x1 − 4x2 +3x3 − x4 =0
2x1 − 8x2 +6x3 − 2x4 =0
� 2.16 Find a basis for, and the dimension of, M2×2, the vector space of 2×2matrices.