Linear Algebra, 2020a

Section III. Basis and Dimension 129
1.42 One of the exercises in the Subspaces subsection shows that the set
⎛ ⎞
x
{ ⎝y⎠ | x + y + z = 1}
z
is a vector space under these operations.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
x 1 x 2 x 1 + x 2 − 1
⎝y 1
⎠ + ⎝y 2
⎠ = ⎝ y 1 + y 2
⎠
z 1 z 2 z 1 + z 2
Find a basis.
⎛ ⎞
x
⎛ ⎞
rx − r + 1
r ⎝y⎠ = ⎝
z
ry
rz
⎠
III.2
Dimension
The previous subsection defines a basis of a vector space and shows that a space
can have many different bases. So we cannot talk about “the” basis for a vector
space. True, some vector spaces have bases that strike us as more natural than
others, for instance, R 2 ’s basis E 2 or P 2 ’s basis 〈1, x, x 2 〉. But for the vector
space {a 2 x 2 + a 1 x + a 0 | 2a 2 − a 0 = a 1 }, no particular basis leaps out at us as
the natural one. We cannot, in general, associate with a space any single basis
that best describes it.
We can however find something about the bases that is uniquely associated
with the space. This subsection shows that any two bases for a space have the
same number of elements. So with each space we can associate a number, the
number of vectors in any of its bases.
Before we start, we first limit our attention to spaces where at least one basis
has only finitely many members.
2.1 Definition A vector space is finite-dimensional if it has a basis with only
finitely many vectors.
One space that is not finite-dimensional is the set of polynomials with real
coefficients, Example 1.11. This is not spanned by any finite subset since that
would contain a polynomial of largest degree but this space has polynomials
of all degrees. Such spaces are interesting and important but we will focus
in a different direction. From now on we will study only finite-dimensional
vector spaces. In the rest of this book we shall take ‘vector space’ to mean
‘finite-dimensional vector space’.
To prove the main theorem we shall use a technical result, the Exchange
Lemma. We first illustrate it with an example.