College Algebra Exam Review 157

College Algebra Exam Review 157 - 167 3.3. VECTOR SPACES...

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Unformatted text preview: 167 3.3. VECTOR SPACES linear map from V n to V s . Namely, if C D .ci;j /, then " # X X Œv1 ; : : : ; vn  C D ci;1 vi ; : : : ; ci;s vi : i i If B is an s –by–t matrix over K , then the linear map implemented by CB is the composition of the linear maps implemented by C and by B , Œv1 ; : : : ; vn  CB D .Œv1 ; : : : ; vn  C /B; as follows by a familiar computation. If fv1 ; : : : ; vn g is linearly independent and Œv1 ; : : : ; vn  C D 0, then C is the zero matrix. See Exercise 3.3.10. Proposition 3.3.25. Let V a finite dimensional vector space with a spanning set X D fx1 ; : : : ; xn g. Let Y D fy1 ; : : : ; ys g be a linearly independent subset of V . Then s Ä n. Proof. Since X is spanning, we can write each vector yj as a linear combination of elements of X , X yj D ci;j xi : i These s equations can be written as a single matrix equation Œy1 ; : : : ; ys  D Œx1 ; : : : ; xn  C; where C is the s –by–n matrix C D .ci;j /. If s > n (C has more columns 23 ˛1 6:7 than rows) then ker.C / ¤ f0g; that is, there is a nonzero a D 4 : 5 2 K n : ˛n such that C a D 0. But then X ˛i yi D Œy1 ; : : : ; ys a D .Œx1 ; : : : ; xn  C / a i D Œx1 ; : : : ; xn  .C a/ D 0; contradicting the linear independence of fy1 ; : : : ; ys g. I Corollary 3.3.26. Any two bases of a finite dimensional vector space have the same cardinality. ...
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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.

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