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# 5.2 - Section 5.2 Properties of Linear Transformations 309...

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Unformatted text preview: Section 5.2 Properties of Linear Transformations 309 5.2 Problems Finding Kernels Find the kernei for the linear transformm tions in Probiems 1—11. Describe the kernel. 1. T : R: —> R2, TUE. y) m (-"L y} 2. T : R3 ——+ R2, Hr. y, z) m (2x + By — Z, -.x +4}! +62) 3. T : R3 ——> R3, TOY. y. z) m (x. y. 0) 4.T:ER3—->IR3. T(x.y,z)=(x—z.xw22.y~z) 5.D:C5~—>C, D(f)=f’ 6.DZIC2—>C. 02(flwf" 7. L1 16‘ —-> C, mm = y’ + 190).)! s. 1:... :c" —> o, L,,(y1 = 31‘") +a1(t)y""” + "” +anml(t)yl +3110}? 9. T : M33 ~+ M32, NA) = AT a b c a O O '1' [d e f m 0 e D g h i 0 O i T(p)=/ pmdt forﬁxedx o 10. I I M33 —+ M33, 11. T:ng——> 1P3, u. Caiculus Kernels The transformations in Problems 12m15 should be familiar from calculus. [dentijjz each transforma— tion and give its kernel. (Problem .14 can have many correct answers.) 12. THEE-«+1133, T(a13+bt+c)=2at+b 13.1“:111’2 -+ 1102, Natl +1.1: +c) = 2a 14. T:iP’2—>l?g, T(atz+bt+c)m0 15. T 1P3 ~+ 1P3, T0223 + at2 + c: + d) = 6at + 2b Superposition Principle For Problems 16—20, suppose that T : V —-> W is a linear Iransfonnatioa ﬁom vector space V to vector; space W. Also suppose that ii; is admiration of Nil) m b1, and that ﬁg is a sohatioamofﬂﬁ) m 112.. Then {il + ﬁg is a solution of T(ﬁ) m b; + b3; this is called the Superposition Principle, asﬁrst introduced in Sec. .2. I. 16. Use linearity to prove the Superposition Principle. 17. Show that y m cost — slot is a solution of the nonhomo- geneous linear equation y” m y’ m 2y = 4sint — Zoos t. 18. Show that y = 1‘2 ~— 2 is a solution of y"—y’—2ym6-2t~2t2. 19. Use Problems 17 and 18 and the Superposition Principle to wn‘to the general solution of y"--y'—2y=4sint—2cost-i-6—2t—212. 20. Genet‘alize the Superposition Principle to three or more terms. Dissecting Transformations In each of Problems 21—40, a transformation TN) 2 Air, 1" : R” ——> R”, is given by a matrix A. For each transformation, ﬁnd the kernel, the image, and their dimensions Determine whether the transformation is infective or .surjective. 00 10 21" [0 O] Rio ~1l 10 12 2.3.[0 0] 24. [4 1] 12 11 25.[2 4] mug} 111 121 37'l121] 23"[24al 121 131 39 [212] 3“[221] 11 12 31.12 32.[24 11 12 0o 11 33.00 34.21 00 31 121 111 35.011 36.121 001 232 121 121 37.241 38.322 111 231 120 110 39.011 40.010 001 000 Transformations and Linear Dependence 41. Let T : IE3“ —-+ R’" be a linear transformation, and let [it], \$3, 13} be a linearly dependent set in R". Prove that the set {Tﬁg ), Tﬁg), Tﬁgli is linearly dependent in ll?!" 42. Let 1" : ER" w> ill” be a linear oanstrmation, and let {\$1, \$3, 713] be a linearly independent set in R". Give a counterexample to show that (TEE), Tﬁg), T620} need not be linearly independent in 1141’". 310 Chapter 3 Linear Transformations 43. Let T : it” —+ YR'" be an injective linear transformation, and let [v], V3, 63] be a iinearly independent set in lFi" Piove that WWI), Tﬁg), T673); must be a iinearly inde- pendent set in 33'". 44. Prove that if a linear transior‘mation T {naps two linearly independent vecto_rs onto a linearly dependent set, then the equation Ni) x 0 has a nontrivial soiution. 45. Consider the transformation T : it”: ma» 113-;2 deﬁned by mm» m ,. For example, if pm m r3 v— 6! + 4, then 4 RPM) W [W1] (a) Prove that T is a linear transformation (13) Find a basis for the icernei of T (c) Find a basis for the image of' T. Kernels and Images Find the kernel and image of each linear transformation in Problems 46—51 rm) 2 AT Ttplmp’ Tl? film ll fl ,\ M 3 a b _ a+~b 49.2”.Mgg a, TL d]_[c+d] 46. T 2 M32 —> M32. 47- I :F3 "9' P31 48. T : M23 —‘r My, 50. TzlPis —:. Tia,b,£,d,e) =(a,0,c,0,e) 51. Taiwan Tr.x,yi=(x+y.0,xw~y) Examples of Matrices Give examples of matrices A in M33 such that Hi} w A)? has the properties described in Prob— lems 52—54 52.“ The IrnU) is the plane 2x w 3y -l- E, = O 2 53. The Im{T) is the line spanned by { [0] O i 0 54. The KeriT} is spanned by { [0] , [i] } l 2 TruelFalse Questions Answer Problems 55—60 true or false, and give a brieferplanation or" counterexan-iple 55. If A is a square matrix, then KCKAZ) = Ki:er True or false? 56“ If A is a square matrix, then ImlAz) = 1mm), True or false? '57, IfA is a square matrix, than Ker(A) = KerlRREF) True or false? 58. if A is a square matrix, then Irn(A) u im{RREF). True or false? 59. If A and B are n x n matrices, titan is it true or false that KerlA + B) = Ker(A) + Ker(B)? l l 69. Im{A) f'orA: [I E is a line in iii? True or false? 61. Detective Work A transfermation T : R4 —> 1&2 is tie- ﬁned with matrix multiplication to be T6?) = A? it is known that the RRE-F of A is l w}. 3 O 0 O 0 1 Determine dim(Ker(T)) and clinnlmﬂn. Is it" one—to- one? is it onto R3? Find bases for the kernel and image 62. Detecting Dimensions Consider the transformation T :llit1 we. ill4 deﬁned by TH) = B? The RREF ofB is I 0 COD 1 0 0 Determine diniﬂierlﬂ) and dimeUD. Is I" one—to— one? Is it onto R4? 63. Stiil Investigating For the transformation T :W‘ ——> R4 deﬁned by T6!) 2 Av, where A has REEF 1 O O O i 0 O 0 l ’ 0 0 0 determine dirn(Ker(T)) and dimﬁrntﬂ) Is T one-tom one? Is it onto R4? 64, Dimension Theorem Again Considertransforrnation T : 1R3 —> R3 deﬁned by T6?) 2 CV, where C has REEF 1—23 0 00‘ 0 00 Determine dim(Ker(T)) and dimCIm(T)) of transforma— tion T, and decide whether it is injective and/0r surﬁective. 65. The Inverse Transformation If T : V ~—> W is an injee~ tive linear transformation, then we can deﬁne an inverse transformation .T“1 : Im(T) ———> V so that, for each a in im{T), r-irn) = v if and only it no m a Show that If”l is an injective and surjective linear transformation Review of Nonhomogeneous Algebraic Systems Express tire general solution for each system in Problems 66—71 as Section 53 @ Eigenvalues and Eigenvectors .311 the sum ofa particular solution and the solution oj'tlze corre- Review of Nonhomogeneous Second-Order DES For each ‘sponding homogeneous system. equation in Problems 78—81, express the general solution of m _ __ the nonltomogeneous DE as the sun: of a particular solution 66" I + y _ 1 67' 3x _ y + 4 ~ m4 (each is a polynomial in t) and the general solution of the 68. x + 2y m 2 69- I H 23’ = 5 corresponding homogeneous DE. 2”” 3"“? 2‘t+43"m5 73,y”+y’—2y==?.tw3 79.y”-—2y’+2y=4tw6 70" x + By — z = 6 71. x1 ~§« 37x3 — 4x3 m 9 2t— y+32=~3 —2.x1-%- .r3+2.s3m—9 80,y”—2y’+y=t—3 81.y”+y=2t —9I1 + lez "w": —3 82. Suggested Journal Entry I The matrix of a linear trans— Review of Nonhomogeneous First-Order DES in each of formation has been transformed to its reduced row echelon Problems 72—7 7, express the general solution oftlze nonhomo~ form Discuss what infotmation about the transformation geneous DE as the sum ofa particularsolution and the general you can obtain by knowing how many pivots [here are and solution of the corresponding ltomogenenous equation. The in which rows and columns they appear“ homogeneous equations are linear or separable; particular solutions (mostly constant} may be found by inspection. 83. Suggested Journal Entry II The rows ofan m x n matrix 7 .F _ m t 7 = w A, considered as navectors, span a subspace of R" called 7‘“ y y w 3 73' y + "y I the row space of At Its columns span a subspace: of l?" 74 y: + _1_ y = l 75. yr + ivy m E; called the column space of A If a linear transformation 1‘ ' t" I- T : R" —-> R’" is deﬁned by m) m as, discuss the 76. yr + {3}, = 39 77. y! + I), m 1 + [2 relationship to "I of the row and column spaces of A 5.3 Eigenvalues and Eigenvectors .S YN OPS l .S: We study spacial vector directions (eigenvactors) and scalar multi— pliers (oiganvalues) associated with a square matrix or with a more general linear transformation, These eigenvectors and eigenvalues are useful both for understand ing matrices (and the associated transfbrmations) and for applying them to a variety of problems Matrix Machine Introductory Example A linear transformation T : R2 ~> R2 is deﬁned by T03) 2 Ali, where Construct a matrix and watch it v 1 2 transform vectors as fast as you move A = 2 *9 . ( I) the mouse. A vector goes in with a " click, and a transf'otmed vector In general, T maps vectos u to a vector No) in a different direction We have pops up. given examples of this in Fig" 53.1, showing I} and T03) on the same diagram. la>rlél=lil FIG U R E S “ 3A General vectors mapped by T61) = Afi. ...
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5.2 - Section 5.2 Properties of Linear Transformations 309...

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