Unformatted text preview: 3.3. VECTOR SPACES 165 Deﬁnition 3.3.18. Let V be a vector space over K . A subset of V is called
a basis of V if the set is linearly independent and has span equal to V . Example 3.3.19.
(a) The set f1; x; x 2 ; : : : ; x n g is a basis of the vector space (over K )
of polynomials in KŒx of degree Ä n.
(b) The set f1; x; x 2 ; : : : g is a basis of KŒx.
Lemma 3.3.20. Suppose V is a vector space over K , and A Â B Â V
are subsets with span.A/ D V and B linearly independent. Then A D B . Proof. Suppose that A is a proper subset of B and v 2 B n A. Since A
spans V , we can write v as a linear combination of elements of A. This
give a linear relation
˛j vj D 0
j with vj 2 A. But this relation contradicts the linear independence of B .
I Lemma 3.3.21. Suppose V is a vector space over K , and A Â V
is a linearly dependent subset. Then A has a proper subset A0 with
span.A0 / D span.A/. Proof. Since A is linear dependent, there is a linear relation
˛1 v1 C C ˛n vn D 0 with vj 2 A and ˛1 ¤ 0. Therefore,
v1 D .1=˛1 /.˛2 v2 C C ˛n vn /: Let A0 D A n fv1 g. Then v1 2 span.A0 / H) A Â span.A0 / H)
span.A/ Â span.A0 / H) span.A/ D span.A0 /.
I Proposition 3.3.22. Let B be a subset of a vector space V over K . The
following properties are equivalent:
(a) B is a basis of V . ...
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