Page 63 - Advanced Linear Algebra

Vector Spaces 47

Theorem 1.7 Let be a set of vectors in . The following are equivalent:
=
:
=
1 : ) is linearly independent and spans .
)
2 Every nonzero vector #= is an essentially unique linear combination of
vectors in .
:
=
:
3 : ) is a minimal spanning set, that is, spans but any proper subset of :
does not span .
=
4 : ) is a maximal linearly independent set, that is, is linearly independent,
:
but any proper superset of is not linearly independent.
:
(
A set of vectors in that satisfies any and hence all of these conditions is
)
=
called a basis for .
=
Proof. We have seen that 1 and 2 are equivalent. Now suppose 1 holds. Then
)
)
)
:
: : is a spanning set. If some proper subset Z of also spanned , then any
=
vector in :c : Z would be a linear combination of the vectors in : Z ,
contradicting the fact that the vectors in are linearly independent. Hence 1)
:
)
implies 3 .
Conversely, if is a minimal spanning set, then it must be linearly independent.
:
For if not, some vector : would be a linear combination of the other vectors
in and so : c ¸ ¹ would be a proper spanning subset of , which is not
:
:
)
)
possible. Hence 3 implies 1 .
Suppose again that 1 holds. If were not maximal, there would be a vector
)
:
#= c : for which the set : r ¸#¹ is linearly independent. But then # is not
in the span of , contradicting the fact that is a spanning set. Hence, is a
:
:
:
maximal linearly independent set and so 1 implies 4 .
)
)
Conversely, if is a maximal linearly independent set, then must span , for
:
:
=
if not, we could find a vector #= c : that is not a linear combination of the
vectors in . Hence, r : ¸ # ¹ would be a linearly independent proper superset of
:
)
)
:, which is a contradiction. Thus, 4 implies 1 .
=
=
Theorem 1.8 A finite set : ~ ¸# ÁÃÁ# ¹ of vectors in is a basis for if

and only if
= ~ º# » l Ä l º# »

Example 1.6 The th standard vector in - is the vector that has 's in all

coordinate positions except the th, where it has a . Thus,

~ ² Á ÁÃÁ ³Á ~ ² Á ÁÃÁ ³ Á ÃÁ ~ ² ÁÃÁ Á ³

The set ¸ ÁÃÁ ¹ is called the standard basis for - .

The proof that every nontrivial vector space has a basis is a classic example of
the use of Zorn's lemma.