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Unformatted text preview: Exercises for lecture._ 1) Determine. which of inhe foﬂowing mapg Einear or £101):
a} T : R2 ~+ 333,?”(3531) :2 + 3y,23cy  Lx) b) T 1&9 ~+ KS, i’“(:1:,y) m [32: ~§~ 29,2: y); c) T : R3 ~+ R3, TL
(.1) T R3 J}, 2&2 Th. 7 w,y,z) : {3: —n 2,3; —§— 231,2]; : mug):{x~%~éy——z,m+ym1);
e) T : C ([071» «x? a
f :r : O as, 1)) m} 0‘3 ,
g) T . C1 (R) % CUR); 3M
11)T:C{[O§1]}«~‘7 T(” i) T : 0W: a 000 m x f” + f + 2; 0;
O
f : f’f, \
\ 2 Evaluate the given 1133618.]? map T at the given point u, (or f (15)): T63; 32;, z) 2 {2.2: m y + 32:, 2.1' ~!~« 2y, ~~3:I: + 103; —— z), u 2 (1, ~2, 3);
c) T : 01w, 1» —+ Tm == f’<fi‘) + m fit) : macs); d if : 0&0; Us; ~> R; m; m 1}? mm, m 2 at; e} T211?” m} R2, 6" 1 2;) Tcr; y) = (is — 3mg: —~ 2;), u (2, W4);
)
) 1 —1 O
Hagan?) : 0 Ml 3
2 3 m}. f) Twig!) ﬂ (217  3132': + :mﬂy}, 12'. = {11*2); 3:1 E‘imi the kernel of the foilowing Hagar maps: ’1" :E w} R3} T(m)y) : (2m + 3y,2y n— 23.9; + y + z)
b) T :32 w} R2, Tang} :2: (z + 3y3222+y) C) T CWR) W? GER), Tm : f’« a Name: Math 2025, Quiz #4 1) Determine which of the following sets is not a vector space. Explain your answer: 8") {{wayyz) 6R3 §$+yFZ$U}A
h) {($1’x2’w3’x4) 6 R10 I531 + $2 — $3 + 5%; r: 4}; {f 5 OWE) 12% f : 0}; d} {it E V I T(9:} m 3;} Where V and W are vectar spaces, T : V w} W is iineaz and y E W. 2) Which of the foliowing maps is not linear. ExpEain your answer:
3.} T : E313 w? W, T{m,y,z) 7» (a: +2y,m—3y}; b}T:]1€3 AR T(w>yj,z) :23+33+z+1;
c) ?=C([a,b}) a 1R, mnwjm dt d) T ; cocoa) w (POUR), TU) J”?
3) Evaluate the given Eineau‘ map at the given vector 2%} T=0°°(R} *> OWE3}, TU} mﬂf’+3fa ﬁx) =3 $2; 0) T :C({0.1}} my 3%, TU) : fol f(m)d2:, ﬁx) : {205(21m). d) E" i *—> T(21722=2‘3) = Z} Jr" 222 +£}2’37 (21,22,233) Z Math 2025, Quiz #4 Nalne: Recail the following facts: 
1) Suppose that V is a vector space, than a subset W C V is a. vector subspace if and only if the following heels: (1} u+wEWforailme W
{2) roEWforailuEWanéaElré—ZE‘. Notice that (1) and (2) say that we can deﬁne addition and scalar multipiication for elements in W. All the
axioms for vector spaces follows then because they are valid in the vector space W. Notice also that {2)
implies (by taking a“ = O) that O E W. This can be used to Show that a subset of V is not a vector subspace. If
i) Q W then W is not a vector Space. But notice that the other implication does not hold, i.e., it does not
foliows form 0 E W that W is a vector subspace. Notice can collect (1) one (2} into one condition: ro~+~so€i/V for ail moew, 733619. (Try to prove that?) 11) A linear map T : V —> W between two vector Spaces V and W is linear if ﬁrm 4: so) 2 ﬁrm) + 32%;) for
all um E V and all 73.9 E F. Taking r a O and v a: 0 (and u and s arbitrary) we get by using that 0 u = 8 and
D + {3 z 0: TUB) = T6) ~11): vT(u) 2:: 0 .
Hence a linear map will always map the zero vector in V to the zero vector in W. But again, thin does not work
the other way, there are kite of maps that map the zero vector in V to the zero vector in W but are not linear {see probelm 2, part d).
Recoil now that we can write any vector in R” as a column vector or as a row vector. If we write the vector x E R”
as a row vector x : {$1, . . . ,xﬂ) and if A n {my} ie a n. x m matrix then we can define a iinear map TA : R” m} rim by matrix multiplication
TA{2:3,...,$,1)2 [m1,...,xn][aéj]
5". TL TL
2 aj1$j, Z ngiij, . . . ,Eajmﬁj]
jml j=1 jzl it is a fact that any linear map R“ w R” can be obtained in this way. For that let (3,;c :2 {0, . i .,G 1 ,.. . ,0). 3
Jam ipiace Then we have
TAléj‘) m (akir  :akm) which Show us that we have to take A to be the matrix with row—vectors T(8j).
We can also View ER" as the Space of column vectors. Then every linear map T : JR“ —> E” is given by a, m x 71
matrix {notice the difference in the order of m and n). This time the connection between matrices and linear maps is given by
f :51 [ 332165ij
{Go} 3 1 = E.
[ 3n l i 23:32; anﬂj Notice that this time we are summing over the rows of A and not the column. Question: Given a. linear mop
T : Eli” —> Rm how can you now ﬁnd the corresponding matrix?
We hove the following two facts: Lemma 0.1. Let T : V ~> W be a linear map between the nectar spaces V and W. Then the set
Ker{T} '2 {u 6 V i T(o) : B}
is a, vector subspace of V. Proof. Let um E KerCT) and r, s E F. Then T(m+sv} :TT(H)+ST(U)
=7~ﬂ+30
WU Hence Tu + so 6 Kong"). [I] 2} Which of the following maps is not linear. Explain year answer:
a} T: R3 > RE, T{m,y,z) 2: (a:+2y,:z:—3y}; b) TAR3 AR, T(a:,y,z}:2x+3y+z+1;
c) T : OHM) —> K :m m I: m at
d} T: 090036) m; mam, Tm : f’f. Seiution:
a.) This is a iiuea; map given by 1 1
T(x;y,z):{w,y,z} 2 —3 . You can also Show this directly!
b) This is wet a linear map because T{G, 0, U) = 1 # 0. c) The integral is linear, so this: is a iinear map (f: rﬁz) + 390!) cit 2 r f: f(t}dt «i» s f: 96} di) (1) This map is not linear. We have T(rf) m (Tf)’{rf) m 2fJ'f ¢ 'rf’f if ff 74 0 and r 7!: 0, *r y’: 2. You can for
example take ﬁx) : x2 and r = 2. Then Tfrf) = 4v{2:c) m2 :83"? and
rT(f} = 2 « (2x} ‘ $2 = 4562.
Notice that Tm) : 0!
3) Evaluate the given linear map at the given vector a) T : cw (R) we» CWGR}, Tm x 2;“ +3f, m) 2 $2; wrwawrawkbi i)<:>=<:>=<:>; c) T:C’({U.1}} m} IR, m) = 1;} mm, m) = mam. d) THC? —+ £13, T{zl,zg,z3) : 21 +222 + {3+i)331 (21,2323) 2 (1 'r'£,1 ~i,2+33’). Seiution:
a) Tm?) = 2~2$+3~322 : 4374—3552. 13} II
momT—Hwﬂd‘lrm—i
!
. ﬁNHL
Mwl—l
Lw—i \ 6:"!
_ egoto. V
1 m1 ‘ 1. Let A 2 (2) (2) ,and deﬁne T :R2 —> R2 by T(x) = Ax.
. l
FindtheimagesunderTofum [43} ande In Exercises 36, with T deﬁned by Tbs) = Ax, ﬁnd a vector x
whose image under T is h, and determine whether it is unique. 1 0—2 —1
3.A='——2 l 6,13: 7
3«2 ‘5 —3
1—3 2 6
4.A= 0.1—4.132 may
3~5 ~9 H9 1 m5 —7 —2
5.A2[m3 7 5],bm["2]
1 m2 1 1
3 4 5 9
6'4” or 1 1 ’1’: 3
—3' 5 —4 ——6 7. Let A be a 6x5 matrix. What must a and b be in order to F deﬁne T : IR“ ——> W by T01) w Ax? 8. How many rows and columns must a matrix A have in order
to deﬁne a mapping from R4 into R5 by the rule T(x) = Ax? For Exercises 9 and 10. ﬁnd all X in R4 that are mappecl into the
zero vector by the transformation x s—> Ax for the given matrix A. 1 I ' m4 7 —5
9. = 0 1 —4. 3
2 "6 6 w4
. ‘ (t=1)q l»! r 4
1 3 9 2 N a .
i 0 3 —4 ,
i. r = 0
10. A 0 1 2 3 ( )p
—2 3 O 5
——l
11. Let b_ m l , and let A be the matrix in Exercise 9. Is is in'
O .
the range of the linear transformation is H Ax? Why or why
not?
_1 5
12. Let b m _i , and let A be the matrix in Exercise 10. Is
4 .
b in the range of the linear transformation x t—> Ax? Why or
why not? In Exercises l3~16, use a rectanguiar coordinate system. to plot 11 = , v = [ _i] , and their images under the given transfor mation T. (Make a separate and reasonably large sketch for each exercise.) Deseribe geometrically what T dees to each vector x
in RE. 22. a. Every matrix transformation is a linear transformation. b. The codomain of the transformation 31: M Axis the set of
all linear combinations of the columns Of A. c. If T :lR" —> lit“ is a linear transformation and if c is in R“! then a uniqueness question is “Is c in the range of T?” “x. c d. A linear transformation preserves the operations of vector
addition and scalar muitiplication. ' e. The superposition principle is a physical description of a
linear transformation.  .5 0 0
2.LetA= G .5 O ,u: ﬁneTziR3 —>R3by T(x)= it 0 0.5 1 a
O , and v = b . De
—4 c ‘ Ax. Find T011) and T(v).
W [‘3 lit] melt Slit] “H3 illiil _ O i x; 011.1
Let T : R2 ——> R2 be a linear transformation that maps
5 . 2 1 . v . —l
11w [2] Into andmapsv.— [3] mto[ 3]. Usethe
fact that T is linear to ﬁnd the images under T of 3:1, 2v. and '
3n + 2v. ‘_ 13.
3.4.
15. 16. 17. 28. Let u and v be vectors in R". It can be shown that the set P of
all points in the paralielogram determined by u and v has the
formau+bv,for0 5a 5 1, 05b 5 1. Let T :R" —> R“ be aiinear transformation. Explain why the image of a point
in P under the transformation T lies in the parallelogram de termined by T01) and T(v). 25. Given v a 0 and p in an, the line through p in the direction of
v has the parametric equation x = p + Iv. Show thatma linear?
transformation : lit” —«> R” maps this line onto another iine
or onto a single point (a degenerate line}. b. The'line segment from p to q is the set of points of the form (t _. ﬁp+ ;q for 0 5 t 5 l {as shown in the ﬁgure é—below) Show that a linear transformation T maps this line
segment onto a line segment or onto a single pomt. 19. Lete1= [(1)],eg=[2],Yi=[§],andyzm:—;].and let T : 3R2 —>~ R2 be a linear transformation that maps e1 into yl and mans a; into yz. Find the images of _§] and . __ x; M —2 __ 7
20.Let x—[xz], v1m[ 5], and vz—[~3], and let T : R2 ——> R2 be a iinear transformation that maps x into
xlvl + x2v2. Find a matrix A such that T(x) is Ax for each x. In Exercises 21 and 22, mark each statement True or False. Justify
each answer. '21. a. A linear transformation is a special type of function.  b. If A is a 3 x5 matrix and T is a transformation deﬁned by
1'01) 2 Ax, then the domain of T is W. c. HA is an m x n matrix, then the range of the transfonnatio‘n
x we Ax is R“. d. Every linear transfonnation is a matrix transformation.
6. A transformation T is linear if and only if T(c]v1+ ; 02m) 2 c1 Tun) + €2T(Vg} for all v1 and v; in the domain
of T and for all scalars c1 and cz. 5 if. 34% i3} 3 “i351; its Elﬁ‘Ws ’4; 'Wwe “ ‘Nt of at"
W ‘. a»
re a ' , E E 4%: i~1eeﬁii,eeza weights? es rm.‘ s i: twee es. WW W
A projection onto the x2 ~axis.
' 3 6 m2 4 13 2xl w x2
. 17. , , 19. ,
3.. , 3. xx 1 ,unique solution {3] i: 6] [9] i 7} [5961 +6ng
2  2.1. Read the text carefuily
H 3 _ Notice that Exercise 21(e} is a
i  5_ x z 1 , net Unique 7_ a 2 5’ b m 5 sentence of the form
0 “(staternent 1) if and only if (statement 2)”
9 7 Mark such a sentence as True if (statement 1) is true
2; t 4 __3 whenever (statement 2) is true and also (statement 2) is true J' x w x3 I ‘1’ x4 0 wheneVer {statement 1) is true. 0 1
11. Yes, because the system represented by [A b} is 25. Hint: Show that the image of a line (that is, the set of
consistent. images of all points on a line} can be represented by the
‘ parametric equation of a line. 13. b. Considerx = (1 ~— Up + tq fort such that 0 5151‘
Then, by linearity of T, for O :5 t 5 l Tm) x ml — r)? + “1) a (I  emu) + mg} (=9 If T{p) and T(q) are distinct, then {*) is the equation for
the line segment between T(p) and T(q), as shown 11]
part (a). Otherwise, the set of images is just the single A reﬂection through the origin ...
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This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.
 Spring '08
 Staff

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