253Solutions6

Linear Algebra with Applications (3rd Edition)

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ma253 ,Fa l l 2007 — Problem Set 6 Solutions *1. Let P be the linear space of all polynomials (of any degree). Show that no ±nite set of polynomials { p 1 ( t ) ,p 2 ( t ) ,...,p m ( t ) } canspanallof P . (Hint: since there are a ±nite number of polynomials in my set, there is one of largest degree.) Let N be the largest of the degrees of the polynomials p 1 ( t ) ,p 2 ( t ) ,...,p m ( t ) .Then no linear combination of these polynomials can have degree greater than N .S i n c e P does contain polynomials of degree greater that N (e.g., it contains t N + 1 ), the set { p 1 ( t ) ,p 2 ( t ) ,...,p m ( t ) } does not span all of P . *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 indepen- dent? Independence tells us that the equation x 1 ~ v 1 + x 2 ~ v 2 + ··· + x k ~ v k = ~ 0 has a unique solution, namely setting all the x i to zero. Now, given how we deFned the matrix A , this is the same as saying that the equa- tion A ~ x = ~ 0 has a unique solution (namely, ~ x = ~ 0 ). That will happen if and only if
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This homework help was uploaded on 01/23/2008 for the course MATH 253 taught by Professor Ghitza during the Spring '07 term at Colby.

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253Solutions6 - ma253, Fall 2007 - Problem Set 6 Solutions...

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