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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: [Scalarmultiplication. 'Elxamples includematrix "multiplication. _Op_er_a'toi:__s_,". integration anddiﬁeren'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. hwysorry, I(A)=AT , . n b __ n b
inningwin, 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 “ivaxis 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'__(frdoing _='_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) (Ow2) 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 xdirection (a) What linear transf‘omatioa matrix would perform a
shear of one unit in the ydirection 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 xdirection 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 iShape in Fig. 5 if} Graph the {sans—
t‘ozmed rshape. 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 lonit 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 [UNV) 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 waistlineS 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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 Fall '07
 RickRugangYe
 Linear Algebra, Vector Space

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