5.1 - 2.94 Chapter 5 mi Linear Transformations mathematics3...

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Unformatted text preview: 2.94 Chapter 5 mi Linear Transformations mathematics3 The blending of linear algebra and differential equations in this text is, in a sense, one of the fruits of their labor: Some Common Linear Transformations Table 513 lists several important examples of linear transiennationsc We have discussed the first four, and wili discuss the last in Chapter 8. The reader is asked to verify the linearity of various linear transfoxmations in the problems. Table 5.13 Common linear transformations I. T : R" mt lit” T62) : ASE Multiplication by an m x :1 matrix A at D” :01 ms (3 mm 2 ft") nth—den'vative operator III. I : C —> R m b Definite integral operator at) — it.er (find in, H) IV Ln : C" ~> C L,I(y) : yi") + a1(t)y("“”+ nth—or‘deriinear ' ' ' + flnmtltly' +~ GNU) y differential operator (continuous a1, a2, a n . , on) V“ E : X --> Y Laplace transfonn £(f) 2/0 e_“f(t) (it (for appropriate spaces X and Y). See Chapter 8, Linear ttansfbrmations T : V —+ W are sometimes called linear operators. If the domain and the codomain are the same (i.e., T : V —+ V), then T is called a linear operator on V, .__ transfonnations functions that map vectot Spaces__j_into -_pt:¢s¢ry'ittg'itie¢ait : addition .- 5an: [Scalar-multiplication. 'Elxamples include-matrix "multiplication. _Op_er_a'toi:__s_,". integration and-difieren'tiation? ._ . 5.1 Problems Checking Linearity For the mapping defined in each of .3. I“ : R2 —s~ R2, Rx, y} = (x31, 23;) Problems 1~16, detemtine whether or not it is a linear trans- formation. 4. 1' 1&2 w) 3R3, T(:c, y} = (x, 2, :c + y) . 2 3 m 1“ .' R? W R“ Tet, m xy 5- T - '_> R s T(x1 "W" (xv 0a 2» T : R2 —> R3, T(x, y) = (x +y. 2y) 6. T1R2 —+ R4, Tlxt 3’) = (-17, it y, 1) 2Hilbert had immensely powerqu insight that penetrated into the depths of a question and made unique connections with sirniiar situations in fimflung mathematicai fields. He made important contributions to many fields, including fonctlonal analysis, integral equations and quantum mechanics His famous list of 2.3 open questions delivered at the Second International Congress of'Mathematlcians in Paris in 1899 showed the vitality of mathematics; many of these probiems were solved in the twentieth century and brought forth new fields and new questions as a resuit Banach was clever at mathematics and attracted the supportive attention of key persons He ioved to work in cafes, either with others or in solitude; he produced fundamental results in topotogical vector spaces, and he also developed a systematic theory of functional analysis Banach published many papers and textbooks, and began a journal and a set of monographs to help pubilsh the work of' others as Well swam”, i an.“ A rt 3»... r r :C[0, 11—» n, rrr) = its) 9“ T:C'[O, 11m» C[O, 1], “rm zrf'm 1o" T:C3{0,1]—>C{D.1], m): f”+2f’+3f 11. rowan], r(ar3+br+c}=2az+b 12. TAPE—Hill, ' T(nr3+bt2~§~ct+d)mn+b 1.3. hwy-sorry, I(A)=AT , . n b __ n b inning-win, TL (ii—C d‘ 15. Int/1123 en, T{A) =Tx-(A) 16. T : R” —> R’", T62) 2 Ai, where A is an m x in matrix 1']. Integration Show that the integration operator 1: C[n. b} ——>~ iii defined by 1: Jill 2/ fwd! is a linear tI‘anstrmatiOn. Linear Systems of DES Show that the .Syslems of linear dif— ferential equations given in Problems 18 and 19 are linear" transformations, where x = x(t) and y m y(t) e C 1( I )r 18. T(x, y) = (.I’ ~— 31, 2.x + y'} 19- Tire. y) = (x + y’. y —— 2x + y’) Laying Linearity on the Line Determine whether or not the mappings in Problems 20—25 are linear trnng‘ormntions from E9. to 3?: (a and b are real constants) 2i}. fix) = 21. T(x) = ox «iw b i . ,. m .3. i z .. 2 22 Hr) ax +5 2 Ht) 1: 3 .x 24. T , = I M 2". I a = W (t) sum 3 (r) 2+” Geometry of 3 Linear Transformation For Problems 26—28, let T : R2 —> R2 be the linear transformation given by Tfi} = AV, where 1 2 A 2 [0 1r . 4er and explain why this means that the .r~axis is mappeé onto itself 26. Verify that Section 5 ‘i ix) Linear Transformations 95 Till = lilil and explain Wily this means that the “iv-axis is mapped onto the line y = x/Z, Tlil = i3l+i2il and use this fact to give a geometric interpretation of the mappings 27. Verify that 28“ Verify that Geometric Interpretations in R2 Construct a marrix repre— sentation for the transformations in Problems 294'], and give a geometric interpretation ofrlie mapping from R2 to R3, Make sketches to illustrate your conclusions. 2.9. 1"(1‘, y) = (x, wwy) 30. T(:c, y) m {.r, 0) 31" T(xi = (x) x) 32“ Composition of Linear Transformations . IThe'conipositionJ ':' iii-_—> Wioftwolinear I - "ransnnnatnstr 2W ands. .U .~+‘ is. . :17 noun or '1 ii- ' 1 .. .._ -. ‘ '~ . _-:-"-fiE:i_..__i'-__(fr-doing _='_irj(s(fi)_). ' 1 f- Show that the composition Ltnnsfbrmation is also a linear transformation Find the Standard Matrix For each linear tronsfornmtion T : iR“ ~> ii?" in Problems 3.3»«40, determine the standard matrix A such that T6?) 2 A? 33. Rx, y) m x + 2y 34. Rx, y) z (y, —x) 35- “I. y) = (x + 2y. x ~ 2y) 36. T(.x, y) = (x + 131,36 *- 2)’, yi 3'7. T(x, y, z) = (x + By, A; ~— Zy, x + y — 2a) 38. T{U1,i)g, v3) = U; + v3 39 Hog, v3, 2);) m (U, + 21);. v3, mu; +4v3 + 3U3) 46. N211, v2, U3} 2 {112, v3. “Uri Mapping and Images For each linear transformation T : R" —> lEE’" given in Problems 41—48, compute the image under T of ii, and find the vectorfs), if any, that are mapped to (V. 41. Hr, y) = (y, —-.r), ii {0, 0), {it 2 (0,0) (1, 0), Ii 42- Tito r) = (I + y, x), 1"! \"V =(3.1) 296 Chapter 5 Linear Transformations I”: #1:)” Th“. y, z) = (I, y -i- E}, 5 m (0. 1,2), V" 2(1,2) 44. T{il],l£3)=-“(ul7“l +2r12}, fi=(1,2). {V={1,3) 45. TUthltz)=(Hg.ll1+li3,t£1—1!1), (1,1), W=(1,1,0) ll f; 46. I{l£l,[lg) =(itg.n1, n; +112), :i ll A W [\J v 5: ii A 5‘.) M w W 47. Nag, M2, :13} m (Hi + “3, “2 W “3). fim(1.1,1).fi’=-=(0,0) 48. Hui, Hg, u3) = (in, “3, ul + H3), 13 =(L2, i}, W={0,D. 1) Transforming Areas For Problems 49—52, let T : R3 —> R3 be defined by TN) m AV, where 1—1 ll 49“ Determine the image under the map of the square hav» ing vertices (0,0), (1,0),(1, l}, and (0, l) . Calculate and compare the areas of the square and its image 50. Repeat Problem 49 for the triangie with verticcs (0, O}, (1.1).andtdl. I) 51. Repeat Problem 49 for the rectangle with vertices (O, 0), (1.0),(i.2),and(0,2). , (A) {2, 2) {0. 0) (LG) (D) (0,0) (E) (0,1) (0, D) (.5. 0) {0. -2) (1, ~11} U! 1‘.) . Caicalate the determinant IAI‘ Can you guess a connection with the resuits of Problems 49—5 1 ? What additionai data might you collect? 53" Transforming Areas Again Repeat Problems 4—9—52 for the linear transformation defined by the matrix at; 1;]. Do the results agree with any conclusion you drew from Probierns 49—52? Can you argue. explain, or prove your coniectare? 54. Linear Transformations in the Flane (G, 2) {0.0) (1,0) FIGURE 5.1.8 The L~shape used in Problem 54. Images of the Lnshape (Fig. 5.1.8) under various transfor- mations of the plane are shown in Fig. 5 .19. Each trans— formation is one of the foliowiag types (A)»~(E): (a) scaling (dilation or contraction}; (b) shear; {C} (O, 1} (1,1) (—2.0) (0.0} (0,0) (F) {0.2) (G) (0.0) {LG} (I, l) (0,0) (Ow-2) Fl (3 U R E 5 . '1 "9 Linear transformations of the Lashape for Problems S4~59 to) rotation; (cit :eflectioa; (e) nonlinear For each image, Specify which type of'transformation pro- duced it. HINT: Consult Tabie 5 i i. Finding the Matrices Each matrix in Problems 55—59 corresponds to one of the linear nomformarimm in Prob» [em 54, Match each matrix with the corresponding image from Fig. 5 J 9. HzNT: Loolc at who: happens to the unit vectors a awn 59. Nu[i 1] 60. Shear Transformation In Example 5, we iookeci at a shear transformation that produced a shear of one unit in the x-direction (a) What linear transf‘omatioa matrix would perform a shear of one unit in the y-direction on the r—shape in Fig 51.6? Which image in Fig 511.10 entresponcis to this transformation? (b) Find the manices {or the other two shear transforma— tions in Figs Silt). 61. Another Shear Transformation The matrix for a shear transformation of 2 units in the x-direction is 12 M=lo t} (A) (B) (0. 0) (O, 0) F I G U R E 5 . “I -1 0 Sheet transfozmations of the reshape for Ptoblem 60 Section 5 '% ill} 62. 63" (C) {0. 0} Linea: Transformations 297 Apply the t: ansibrmatéon malt ix M to the matrix shown in Tobie 5 1.2 iGI‘ the i-Shape in Fig. 5 if} Graph the {sans— t‘ozmed r-shape. Clockwise Rotation In Example 6 we looked a: a coun- terclockwise rotation about the origin Write the matrix for a 30" clockwise totation of the Oi'igina] {shape Graph the tz'ansi'onned r—shape Pinwheel The pinwheei in Fig 5 H] is obtained from the r—shape (Fig. 5.16) by a shear transformation Di —1 units in the y~diiection foiiowed by a successioa of 30" rotations (We are assuming that each successive transibr— motion leaves a “print” so that the end result is the pinwheei shown.) (:1) Determine the matrix for the shear transformation and the number n of successive rotations required to com— plete the pinwhecl (b) Is it true that (R3D=)" = I f01 some :1? Explain HGU RE 51 .i 1 Piuwheei (scaled) fon‘Ptoblem 63 293 Chapter 5 ill Linear Transformations 64» Flower Expiain how the flower in Fig 5‘ l ‘13 can be ob— tained from the F~shape in Fig 5.1 12 Describe the sucm session of matrix transformations C=(D,2) 13:0,?) mg Emmi} Dr(l.l) Am{0.0) Fl G U R E ‘3 ”1.1 2 The F—shape used in Problem 64. FIG U R E 5 .1 .1 3 Flower (scaled) for Problem 64. (is. Successive Transformations Recoil that linear transfor- mations can be applied in succession by composition (de- fined in Problem 32). The conceponding process for linear transformation matrices is matrix multiplication. Consider a l-onit shear in the y—direction followed by a counter- clockwise rotation of 30", Find the transformation matrix formed by the product of' the two matrices. Sketch the transformed r—shape 66" Reflections (a) Reflect the reshape (Figt 5716) about the .rvaxis and then about the y—axis. Find the transformation matrix M=&m, where Rx is the matrth for reflection about the x~axi5 and Ry is the matrix for reflection about the yards, and sketch the transformed r—shape (b) What counterclockwise rotation is equivalent to these successive reflections? Illustrate with a sketch, . Derivative and Integral Transformations in the vec~ tor space CW [(1, b] of infinitely differentiable functions on the interval [(1, [1], consider the derivative transformation D and the ciefinite integral transformation I defined by CS? ‘5 DifXA‘) m fit) and [UN-V) m ftfflit (a) Compute (01}(1‘) = D(I(f}). (13) Compute ([DKI) = “DUN. {c) Do these transformations commute? That is to say, is it true that (D1)(f) = (ID}(f) for aii vectors 1‘ in the space? 68. Anatomy of" a Transformation The iinear transforma— tion I : R2 —> lit4 is defined by T5!) = Av, where 1 wt 1 O Ams 1 1 O (:1) Determine the vectors in R3 that 1' maps to the zero vector in 11%“. (b) Show that no vector in R2 is mapped to [1, l, t, i} in R“ (c) Describe the subspace of R4 that is the image of i" {that is, its range). 69. Anatomy of Another Transformation The linear trans» formation T 1 R3 —> R2 is defined by TW) m Bi", where 1 1 —1 H2 2 4,]. (a) Determine the vectors in 1&3 that T maps to the zero vector in R2. (b) Find the vectors in R3 that T maps to [1, l] in R3, (c) Describe the image (range) of the transformation T. Functionais Mappings from a vector space to the real num- bers are waistline-S calledfimctionals?’ Determine whether the transformations in Problems 70—73 are linear frmctionals from C [0, i] to ER, ' D l 70. Hf}: W 1 71- Tif)=/ ififlidt .0 3Referring to a numericaivvalued correspondence defined on a set of functions, the intended sense offimciionai was originain “function of a function 7‘ <1 72. nee—2]0 from 1 7.3” no = / find: 0 Further Linearity Checks Versz that the mappings in Prob- lems 74—76 are linear trong’ormotions 74. L; 35; "9' C. L10”) = y’ -§- p(t)y (p continuous) 00 €~fl fwd: a (X and Y appropriate spaces) 75" £zX—eYyL'tfiw 76. L : X ——> R, L(a,,) = iim a,r JIM->00 (X the space of convergent reai sequences) Projections Use the definition of projection, as stated here, for Problems 7 7~80. I.Projcction_"3_j".._ .' _ ._ . __ _ __ __ 5A linear transfonnntion IT : V ——>_ W? where W isa sub— ._ _ ' ;_ space of V, isicalled a projection provided that}; when ' _ restricted to W, reduces to the identity mapping; that is, .' -. -_ flit) for._ali_vectors_ in the subspace j}; ._ - ' 77. Verify that the transformation in Exampie 1 is a projection, What is W in this case? ’78. Verify that the transformation in Example 2 is a projection What is W in this case? 79. Explain why the transformation 1' : R3 m} R3 given by Tu, y, z) 2 (WI, 0, 3.1:) is not a projection. Identify sub— space We 80" Is the iinear transformation T : R3 ——> R3 defined by TU, y, z) = (x + y, y, 0) a projection? Explain 81. Rotational “Transformations A mapping T : R2 ~> R3 is given by N?) = Av, where cos 6 A: [new —sm6l:l‘ smB Show that T rotates every vector?! 6 R3 counterclockwise about the origin through angle 6. HINT: Express v using Section 51 :33 Linear Transformations 299 polar coordinates, o [1" cos a] v = . , r sm or and use the identities for cos(8 + or) and sin(6 + or) 82. Integra! Transforms If [(65, t) is a continuous function of: andt on thesquareO _<_ s s 1, 0 g I 5 I, and fit) is any continuous function of t for O 5 t 5 t, we can define the function F given by [ F(s)=/ minnow - 0 Show that the mapping Itf(t)) = H5) is a linear trans— formation tr‘om C [0, 1} into itself Computer Lab: Matrix Machine USE the matrix machine from [DE (Lch 1.5) to analyze each transformation _ from R2 to R2 in Problems 83—88, and answer the following questions- (21) Which vectors are not moved by the transformation? (b) Which nonzero vectors do not have their direction changed? (c) Which vectors do not have their magnitude changed? (d) Which vectors mop onto the origin? This set is called the nullspace of the transformation (6) Which vectors, ifony, mop onto [ .7 (f) Find the image of the transformation, and state whether it is all of R3 or" a subset. Matrix Machine $74 Enter a matrix, then point/stick to choose or change a vector; simultaneously you wilt see its transformation by the vectoc 0 i 1 1 83 [4 D] [1 1] .. 0 l i ~2 8;, {I 0] [fig 3] 2 0 1 2 [a 3] [I O} 89" Suggested Journal Entry Give an intuitive descrip~ tion of the difference between a linear and a nontinear transformation. Do you find that your impressions are more aigebtaic and computational or more geometric and pictorial? ...
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This homework help was uploaded on 04/07/2008 for the course MATH 5A taught by Professor Rickrugangye during the Fall '07 term at UCSB.

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5.1 - 2.94 Chapter 5 mi Linear Transformations mathematics3...

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