# Linear Algebra with Applications (3rd Edition)

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Unformatted text preview: 4.3 .10 0100 00 m (rum. 2.1,etBbetl1ebasls [0 0].[0 o].[1 o],[0 1]ofR .Thenthecoormavs l l 1 l 1 1 2 _ 2 2 3 the given matrices with respecthnre l 1 B = 1 . 3 4 B— 3 . 5 7 B- 1 2 l l l 2 1 l U l} —l 3 1 4 4 . . 1 2 3 4 U l U 4 5] .[6 3L — 6 . Finding Rd 1 3 5 6 — U u l _1 7E I4, we ?' 8 l 4 7 8 U 0 0 0 l l 2 l 1 2 3 4 . muclude that the four Vumors 1 , 3 , r, . 6 are hnearly dependent. and so are l 4 T H . . l 4 1 1 l 2 2 3 the four given matrices. In fad. Li 8] - — [I 1] + 4 [3 4] — [5 7]. 48. Use a diagram: aoos(t)+bsin(t) T bcos{f)—asin(t] l J [*3] T [—3] 0 l ThusB— [_l 0]. 5|). B = [[(b — l] 0050) — usi11(£)]a [at-05(1) + (b — l) sinUﬂB] = [it-a] bf: I]. 52. Recall that. was“ -- 6) = cos(6)cos(r) + sin(6) slum and “mu — a) = cwMJSinU) - si:1(§)ms(f). Also, coshr/4) = 51mm) = Jim. Thus 3 = [[wsu — arr/4)]; [sh-(r- — «x4113! = [[sgicostt} + -!?siu(t)]a [—’-?°03m+ jgﬂwﬂsl = a“? -1] U —1 ll 66. a. B: [[T(Jt‘f)]B [T‘IIIQHB [THEHB] = [2 U ~2J. U l {J l U —1 0 —l I). Note that Ire-[(8) = 0 1 I) . Now im(B) = spam 2 . {l and 0 0 U 0 l l ker(B) = span 0 . Thus 23.1.22. —xf + is a basis of im(T) and If + 4:3 is a basis 1 of kerﬂ‘). 0.1. 11]. fi- 1? = 2 + 31.- + 4 = 6 + 3.1-. The two Vectors enclose a right angle if E- ii = 6 + :11; I that is. in: = —2. 16. You may be able to find the solutiom by educated guming. Here is the systematic approach: we ﬁrst find all vectors .i-' that are orthogonal to 51,53, and 63. then we identify the unit Wetors among them. Finding the vectors 53' with z" - 5'. = E - 12 = .ir' - (I; = U amounts to solving the system n+r2+xs+n=0 x.+zg—:3—.r4=0 ail—12+I3—qu=0 (we can Inuit all the coeﬂicienls {n f . .. 272 —f The solutions are of the form .1: = \$3 = _t . I; I Since ijll = 2m, we have a unit vector if t = i or t = —%. Thus there are two possible choices for 174: 1 -1 2 '2 _\$ d % nu . “i i _|_ I 2 'i 18. n. lffﬂ2 = 1+§+ﬁ+é+~ - . = i—jr = i (use the formula for a geometric series, with n = so that llfﬂ = a: 1.155. I). Ifwe lot II: (13.0....) and t7: (1%. then 9 = arems 317"" = met.ij = = ﬁ(= 30°]. c. .i'.‘ = (I, 7'5. - - - . 7%, - - does the job. since the harmonic series 1+ & +3'§+- - - diverges (a fact dim-Imam] in intnniuetory calculus classes]. d. Ifwr- let 5: (1.0.0....}.£= (1. g.§.---) and a: ﬁll = a? (14.5") then projﬂ: (s- as: 5‘, (1.1. }.---). 28. Sinoe the three given vectors in the subspace are orthogonal, we have the orthonormal! Imsls l l l .. l .. l _. -l ul=% l 1)::3' _l ru3=é _l l —l I Now we con use Fact 5.1.5, with f = E", :projvf =01] wail + (172 - 5):}; + (53 - 53173 = 3 . 1 5 -1 1 29. By the Pythagorean theorem (Feet 5.1.9). "in? = “7s. _ 3s, + 2.13, + s. — in“? . = Il'i'ﬁ'lll2 + "352112 + “2'53"2 + Ill—114“2 + I155||2_ = 49 + 9 + 4 + l + l = 64. so that ||;i"|| = 8. 40- “'52” = V172 "172 = M02 : 3. 42- “51+52H= x/(ﬁ'n +52) - (6: +172 = m. 46. Write the projection as a linear combination of if; and {£2 : (-161 + ago). Now we want 1'33 i on?! + C252 to be perpendicular to V. that is, perpendicular to both 51 and 133. Using dot products, this boils down to two linear equations in two unknowns, 11 = 3c1 + 5c; and 20 = 561 + 902. with the solution C; = -%,02 = Thus, the answer is —%'t71 + ...
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