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# hw3 - C C is a ﬁnite set of indepedent vectors from U For...

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Homework Assignment 3 - MAS 4105 - Fall 2002 NAME: DUE : Monday, October 7 at the beginning of class. An extra point is awarded for submitting homework on time. This assigment will not be accepted after 5PM October 8. Instructions : Write in complete sentences using mathematical symbols where appropriate. Your reasoning should be clear. Write in such a way that the average student in the class can follow your work. It is OK for you to discuss this homework assignment with anyone, but you must acknowledge any assistance. It is NOT OK to copy anyone else’s work and it is NOT OK to copy from a book. Complete : On this assignment I obtained assistance from Date: Signed: 1. #12, p.120 2. #9, p.132 3. #14, p.144 4. Prove or disprove. (a) { p P 3 : p 00 (0) + p (0) = 0 } is a subspace of P 3 . (b) { A R 2 × 2 : A is singular } is a subspace of R 2 × 2 . 5. #6, p.155 6. BONUS Let U be a subspace of a finite dimensional vector space V . (i) Prove that U is finite dimensional and dim U dim V . Hint : Consider two cases: U = { 0 } and U 6 = { 0 } . In the second case let C := { C : C is a finite set of indepedent vectors from

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Unformatted text preview: { C : C is a ﬁnite set of indepedent vectors from U } . For a set C , let | C | := number of elements of C . Show that C is nonempty and that m := max C ∈ C | C | exists. Hence there is a C 1 ∈ C with | C 1 | = m . (Note: C 1 is a maximal independent set of vectors from U .) Show that C 1 is a basis for U , and use this to show that dim U ≤ dim V . (ii) Prove that if dim U = dim V then U = V . 1 7. Let U , V be subspaces of a vector space W . (i) Prove that U + V := { u + v : u ∈ U, v ∈ V } is a subspace of W . (ii) Now, suppose that U and V are ﬁnite dimensional and that U ∩ V = { } . Prove that dim( U + V ) = dim( U ) + dim( V ) . Hint : Consider two cases: (1) U or V = { } . (2) Neither U nor V = { } . In case (2) let { u 1 ,u 2 ,...,u n } be a basis for U and { v 1 ,v 2 ,...,v m } be a basis for V . Show that { u 1 ,u 2 ,...,u n ,v 1 ,v 2 ,...,v m } is a basis for U + V . 2...
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