Linear Algebra with Applications (3rd Edition)

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ma253 , Fall 2007 — Problem Set 6 This is a short problem set to keep you busy until Wednesday. There are only three problems, and all of them are starred. This assignment is due on All Hallows Eve, Wednesday, October 31 . *1. Let P be the linear space of all polynomials (of any degree). Show that no finite set of polynomials { p 1 ( t ) , p 2 ( t ) , . . . , p m ( t ) } can span all of P . (Hint: since there are a finite number of polynomials in my set, there is one of largest degree.) *2. Suppose we are given a set of vectors { v 1 , v 2 , . . . , v k } in R n . Then we can create an n × k matrix by using the vectors v i , in order, as the columns. Call that matrix A . a. What property of A will guarantee that { v 1 , v 2 , . . . , v k } is linearly inde- pendent? b. What property of A will guarantee that Span ( v 1 , v 2 , . . . , v k ) = R n ? c. What property of A will guarantee that { v 1 , v 2 , . . . , v k } is a basis of R n ? Explain your answers! *3. (This is closely related to problem 58 in section 4 . 1 , and to Example 1 in that section.) A differential equation is an equation that gives a relation
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