Unformatted text preview: n Deﬁnition 3.3.23. A vector space is said to be ﬁnite–dimensional if it has a ﬁnite spanning set. Otherwise, V is said to be inﬁnite–dimensional . Proposition 3.3.24. If V is ﬁnite dimensional, then V has a ﬁnite basis. In fact, any ﬁnite spanning set has a subset that is a basis. Proof. Suppose that V is ﬁnite dimensional and that S is a ﬁnite subset with span .S/ D V . Since S is ﬁnite, S has a subset B that is minimal spanning. By Proposition 3.3.22 , B is a basis of V . n Let V be a vector space over K . Represent elements of the vector space V n by 1 –by– n matrices (row “vectors”) with entries in V . For any n –by– s matrix C with entries in K , right multiplication by C gives an...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08