253Solutions7

Linear Algebra with Applications (3rd Edition)

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ma253 ,Fa l l 2007 — Problem Set 7 Solutions 1. Problems from the textbook: a. Section 3 . 3 , problems * 18 ,* 22 ,* 26 ,* 28 ,* 30 ,* 36 ,* 38 . *18 Well, if el professor is going to ask you to do things involving nasty matrices, it’s nice when they come pre-row-reduced, eh? Reading off from the matrix, the image is R 3 ,soabas isistheusua l e 1 , e 2 and e 3 .Thekerne l is ker ( A )= Span 1 0 - 5 1 0 , 2 1 0 0 0 . *22 Rats ,th ist imewehavetorow-reduce. rref ( A )= 10 - 6 01 5 00 0 , so a basis of the kernel is the single vector 6 - 5 1 and a basis of the image consists of the two vectors 2 4 7 and 4 5 9 . *26 For part (a), start by ±nding the kernel of C :i ti sthespano f 0 1 - 1 . Now just ±nd which of the others have this vector in the kernel: L is the only one. (Notice that this vector is in the kernel if and only if the last two columns are equal, so this is easy to check!). Now row-reduce L to make sureitskerne lisa lsojustthespanof
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For (b), start by noting that the image of C is the span of 1 1 1 and 1 0 1 .I t is not hard to see that this span is just the set of all vectors in R 3 such that the ±rst and the third coordinates are equal. Now it’s easy to see that H and X have the same image as C . For (c), we can eliminate C , H ,and X at once, by part (b). Now notice that T and Y have the same image: both images contain 1 0 0 ,andthefactthat 1 0 0 + 0 1 1 = 1 1 1 shows that they are the same. So L is the matrix whose image is different from that of all the others. *28 Since there are 4 vectors, they will be a basis if and only if they are independent. Now notice that if α 1 0 0 2 + β 0 1 0 3 + γ 0 0 1 4 + δ 2 3 4 k = 0 0 0 0 , we can read off from the ±rst three lines that α =- , β =- , γ = - . Plugging those into the fourth line, we get 0 =- - - 16δ + =( k - 29 ) δ. So the vectors are dependent (i.e., there is a non-zero choice for δ that makes the equation true) if and only if k = 29 . For all other values of k ,the vectors are independent, hence are a basis of R 4
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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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253Solutions7 - ma253, Fall 2007 - Problem Set 7 Solutions...

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