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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’—2ym62t~2t2. 19. Use Problems 17 and 18 and the Superposition Principle
to wn‘to the general solution of y"y'—2y=4sint—2costi6—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" counterexaniple 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 RREF 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 onetom
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), rirn) = 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 SecondOrder 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 FirstOrder 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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 Fall '07
 RickRugangYe
 Linear Algebra, Vector Space, linear transformation, Properties of Linear Transformations

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