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Linear Algebra with Applications (3rd Edition)

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Math 417 homework 5 solutions Section 3.4 Problem 22 A = bracketleftbigg - 3 4 4 3 bracketrightbigg , S = bracketleftbigg 1 - 2 2 1 bracketrightbigg (the columns of S are the basis vectors). S 1 can be computed either by elementary row operations, or by formula which is faster here: S 1 = 1 1 · 1 - ( - 2) · 2 bracketleftbigg 1 2 - 2 1 bracketrightbigg = bracketleftbigg 1 5 2 5 - 2 5 1 5 bracketrightbigg . Thus, by Fact 3.4.4, B = S 1 AS = bracketleftbigg 5 0 0 - 5 bracketrightbigg . Section 5.1 Problem 17 A vector vectorx = x 1 x 2 x 3 x 4 is perpendicular to W if it is perpendicular to each vector in it: For all vectory W , vectorx · vector y = 0 . But obviously it is sufficient to check that for the two given vectors spanning W . (Reason: let W be spanned by y 1 , . . . , y k and assume that vectorx · vector y j = 0 for all j = 1 , . . . , k . If vector y W too, then vectory = c 1 vectory 1 + . . . + c k vectory k for some real numbers c 1 , . . . , c k because the vectory i span W (it does not matter whether they are actually a basis). Then vectorx · vector y = vectorx · ( c 1 vectory 1 + · · · + c k vectory k ) = c 1 vectorx · vectory 1 + · · · + c k vectorx · vectory k = 0 1
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because vectorx · vector
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