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math103hw8solns

math103hw8solns - Maw(03 BMW So‘dJﬂM 6 I 42 We ﬁrst...

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Unformatted text preview: Maw (03 BMW So‘dJﬂM 6.\ I 42. We ﬁrst. expand across the fmlrth row to Obtain 1 0 the second row to obtain 3 det U , then across . . 6 While" wit-HIGH = 6(24 + 10 —12)= 132. o. “"55: (neon: GOP 1 3(2) (let 2 3 4:1. dehﬂcA) -—- Is" deth} 54. a. We will expand down the ﬁrst column: a u o o u 1 0 [in ; det[Mﬂ} a 5det(Mn_1} — Idet MM = 5det(fL-In_1] — 1(6) detfﬂd’,,._2}. H 5 b. d1=2‘d2 :det =4‘ - _ 11 —'I C d — 2'“ The base case has already been shown. If we assume that (ﬁnd - 2 and ,. .n _ . L— r 11- » —1 = - if“ 2 = 2*3'2, then (in = 5dﬂ_1'— ﬁd,,...g = 5(2"“‘} — 6(2’ 2) 2 01:2 1}I—— 3(2” ) 2(291—1) : 2e 40. By Exercise 38, dethTA} 2 {det[AJJ‘2. Since A is orthogonaLL ATE-1 : 1 = detlﬁfn} = det— (ALTA) = [dam]? n SO and det(A) .-_ \$17 42. detIA'rA) 2 dammit“ng : deHRTQTQ-R) = dams—1mm __ dctfRTRJ : detmijdetfm T T T Deﬁnition of A Since columns of Q are orthonormal Fact 6.2.4 VT). 2 = Ida-rm)? > 0 i=1 T T Fact Since R 6.2.? is trjmlgular. 48. Since 5'2, . . . ,1?“ are In: early independent thj = 0 only if :E is a linear con ‘5}- ‘5, {otherwise the matrix [:55 9'2 - ~ 1 ﬁn] is invertible, and T97} aé 0). Hen T is the span of 732, . . . ,ﬁm an (n — l}—dirneneionaJ subspace of IR" real line R (since it must be l-dimeneiunal}. ubination of the. ca, the kernel of . The image of T is the TM ’— 2. We know A17: A630 '5': 14—1111th Adlai: AA‘lﬁ', so 17-“: AA‘IEor 3141?: %5_ Hence if is an eigcnvector of A-1 with eigenvalue 6. Yes, If A1? = M? and 31?: p.13, then ABE? ——n Amt?) = 1:.(At—J’] = JAM? 18. Any nonzero vector in the plane is unchanged hem: I 1. Since any nmmaﬂ Vector in VL e 15: an Elgflllvgctol. with the _ r ei envalue Is ﬂipped about the origin, g ejgunvalue =1 Pick an - it E ml Bigeil‘ilecnor ‘ j . ' 5’ two non-c - . . r “Flt! ham “ﬁnishing of eigenvecmrs' Ullmeat vectors [10111 if and one from V L to form a. l. 42. We will do this in :3. Slightly simpler manner than Exercise 40. Since A ID is simply,r the I] ﬁrst column of A, the first column must be a. multiple of :31. Similarly, the third column must be a multiple of 93‘ There are no other restrictions on the form of A, meaning it can a f; 0 1 0 0 [j b A - -- . 1 U 0 e emiymatnxofthefolm g 3 0].:6 0 U 0 +5 0 0 0 41 0 1 3 + i“ 0 U 0 0 n a I] U 0 9 0 9 0 e e ff 0 0 0 +3 I] 0' ﬂ 0 J. [l 0 CI 1 1 0 0 ﬂ 1 {J Tllue,abasieoflfjs 0 0 [ll 0 0 D 0 {1| 0 U 0 0 It) 0 ' U 0 1 0 , 0 0 0 e n o . 0 0 0l 0 0 0 o 0 0 n ’ and the dimension of V is 5. 1 a D G 1 ...
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