168 3. PRODUCTS OF GROUPS Proof. It follows from Propostion 3.3.25 that any basis of a ﬁnite di-mensional vector space is ﬁnite. If a ﬁnite dimensional vector space has two bases X and Y , then j Y j ± j X j , since Y is linearly independent and X is spanning. But, reversing the roles of X and Y , we also have j Y j ± j X j . n Deﬁnition 3.3.27. The unique cardinality of a basis of a ﬁnite– dimensional vector space V is called the dimension of V and denoted dim .V / . If V is inﬁnite–dimensional, we write dim .V / D 1 . Corollary 3.3.28. Let W be a subspace of a ﬁnite dimensional vector space V . (a) Any linearly independent subset of W is contained in a basis of W . (b) W is ﬁnite dimensional, and dim .W / ± dim .V / . (c) Any basis of W is contained in a basis of V . Proof. Let Y be a linearly independent subset of W . Since no linearly independent subset of W has more than dim .V / elements, by Proposition 3.3.25 , Y is contained in linearly independent set B that is maximal among
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