Page 133 - A Course in Linear Algebra with Applications

5.1: Existence of a Basis 117

as well as the standard basis

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And one can easily think of other examples. It is therefore
a very significant fact that all bases of a finitely generated
vector space have the same number of elements.
Theorem 5.1.6
Let V be a non-zero finitely generated vector space. Then any
two bases of V have equal numbers of elements.

Proof
Let {ui,U2,... u m } and {vi,V2,..., v n } be two bases of V.
,
Then
V = < u 1 , u 2 , . , u m >

and it follows from 5.1.2 that no linearly independent subset
of V can have more than m elements; hence n < m . In the
same fashion we argue that m < n. Therefore m = n.

Dimension
Let V be a finitely generated vector space. If V is non-
zero, define the dimension of V to be the number of elements
in a basis of V; this definition makes sense because 5.1.6 guar-
antees that all bases of V have the same number of elements.
Of course, a zero space does not have a basis; however it is
convenient to define the dimension of a zero space to be 0,
so that every finitely generated vector space has a dimension.
The dimension of a finitely generated vector space V is de-
noted by
dim(V).

In fact infinitely generated vector spaces also have bases,
and it is even possible to assign a dimension to such a space,