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Unformatted text preview: F05 13419 ﬁ‘naﬂ:89€u7h€in 1. For a. nonhomogeneous system of 20 equations in 19 unknowns, answer the following three questions:
0 Can the system be inconsistent? ”(ES H
0 Can the system have inﬁnitely many solutions? YES
0 Can the system have only one solution? (as go A. b
A. Yes, Yes, No. ‘
B. No, No, Yes. _
Wei .5 {q 1, a [a
C. Yes, No, Yes. 0 S A 5 ﬁx
of
D. No, Yes, Yes. CM M 9" W3” 99:1,
@Yes, Yes, Yes. av wf' "A (5/9 a???
.0, 0,0. 1/ aA< “D‘Héj
39 met“. 4
a b c 2a — 39 g d 2 a g 5i a 3.
2 Ifd E f=13,ﬁnd 233—3}? h 8.: 5119 A g :2 in h e
g z 20 — 3: a f 9 C L. f C L ~f
A. 13
B. 26 : a a b C a In C
C 26 _ 
O a h . * 9 a e f
D. 39
d 9 ‘F a [a 4
E. 39
F. 13 11—1
3. LetB: 0 1 1 .Then the second row of 8‘1 is:
O 0 1
.[01'1] ‘4‘y[oo ‘6 (:9wa
1341101 OIIO‘O Noloool'
'C.[011] 00100‘ 00‘
1341—10}
E.[101] F. None of the above 4. If '01 = (1,3,4),1}2 = (2,6,8),vg = (4,1,5) and v4 = (—2,5,3), which of the following is a.
basis for U = span{v1,v2,vg,v4}? n— A.{v1} Leg,1“ 1242 I 24—;
.{01, ’03} 3 6 I .I 5 N C) O "I, H
C.{‘U2,1}3, 94} I g 5"ll 3 0 O ‘1' H
2:1)“ ”2’ ”a; V4 Vs F [ © 1 1+ — 1
 ”I! ‘03, U4 0 O i __t
F {”11 ’02, '03, ”4} O 5. Let A = [a ‘9‘]. Find all ((1,110 for which A2 2 0. Actual) A2: “a 4L a 4 L4 “Milo
0%”) [1 JD 5] '[amag 522; E D. ::(3,—3) W a = 4 E.i(3,3) ﬂ=—b \ Q22 )bcq.
 lol — '1 \a a F. There is no such (a,b). _ Li" W a — .. 6. Let M22 denote, as usual,the vector space of real 2 X 2 matrices and let J = [1 0 ] . The
dimension of C = {A 6 M22  JA 2 AJ} is: :3? [31??? {3112] we D; 3
E. 4 f, M '6‘ CW [aha] 8 —3
18 —7
some invertible matrix P? 7. The matrix A = [ ] is diagonalizable. Which of the following could be P'lAP for U
l—ﬁ
OI—I
I—ID
I—l
E11
l—I
OM
COD
I_._.l
111
[—1
OJ: lo
(:1
I—a—u IFA‘ >ll" le) *3 .. AS’é +54. 7:. 23"“)“2 8. Which of the following are subspaces of M22? (1) The set of all symmetric 2 X 2 matrices (i. e. when A: A‘). 1/ (1)“; [t [ZEIQ’EE d (2) The set of all antisymmetric 2 x 2 matrices (i. e. when A— 4* —A‘) “(2) 1. [301'] la: Q Q fr
0 (3) The set of all invertible 2 X 2 matrices X 44 (“go“) (4) The set of all 2 x 2 matrices with trace 0. (Recall that the trace of [a b] is a. + d.) 1/ (106% «a»: 06m) 6 d
A._(1) and (2)
B. (1) and (3) m ' “Mi: El! [.0 J] Bﬂl’w ‘ ”194“”
C (1), (2) and (3) z .
D. (2), (3) and (4) (7/) :Mll [3 ill ’30 "‘ “ ‘3 9M“
E. (3) and (4) C3) .' I,L 1:. w M — L a m. _ .
@(1), (2) and (4) M 0= 1; +{12,)4‘r ‘ , t m‘f‘«“C'5l3»/w+do¢al
W ”(130? A»? 44);. M 011: amt C2) .1 «WMJ: Wt;
(q) WLWWmL aim (aim CAMJLJ SL1: {HQIMMM spw{[~;°) £23)st 9. Let F[R) = {f  f : R —> R} be the vector space of all realvalued funtions of a real
variable. It is known that {sin 2:, cos 3:} is linearly independent in F[R], and that sin(m + a) = cosmsina +sinxcosa, for 3.11:3,151 E R. “W .What are the dimensions of V 2 span{sin 3:, cos :12, sin(:1: + 1)} and W = Span{2sin (1:, 3 cosm}? (5 4m. was; 24:
A.dimV=3,dimW=3 U 91"“me '2 84nd) ODSX—f— was“ ﬁpwla'nxlcny}
B.dimV=2,dimw=3 C. dimV=3,dimW=2 “(LAM \I . 919% fﬁm}, 606K}, 3‘4“ NM. W?)
E. dimV=1,dimW=3 _ . . F.dimV=1,dimW=2 HOW: Wrmlla‘m)‘;$wvvq Q’Ml'ﬁ": :1.
*1:
P. 10. Suppose A is an 'n, X 5*: matrix. Among the following statements, which one is not equiva
lent to the others? * A. A is invertible. B. Ax = 0 has a nontrivial solution a: (3 R”. M +64 W ”Luff 4?va
ab "FGk o C. A11: 2 b has a unique Solution :1: ES Rfn for every 3) E R”. . D. The determinant of A is not zero. ‘
E. A is rowequivalent to the identity matrix. F. The rank of A is n. 11. (3.) Consider the linear system a: + z = —1 2:1: — y = '2,
y + 22 = —4 (1:1: + 61; + 4.76 = 0 Find all a andc so that this system has (i) a unique solution,
(ii) inﬁnitely many solutions, and
(iii) no solutions. . , MEA‘Q‘ Y 5'“
(0% Go’stew/ W“ M 1 :3; .“f
a swung» ¢=> c #0 [MM mime) % 11. (b) Consider the network of streets and intersections below, known as a ‘roundabout‘. The
arrows indicate the direction of trafﬁc ﬂow along the oneway streets, and the numbers refer to
the exact number of cars observed to enter or leave the intersections during one minute. Each 3:;
denotes the unknown number of cars which passed along the indicated streets during the same
period. Write down a system of equations, together with all constraints, that will determine all
possible trafﬁc ﬂows. DO NOT SOLVE THIS SYSTEM. C(ucﬁmwﬁs .’ X4 ? 0 c‘:{).., 6: 2 x," € 2!
3734‘“— W ”U 1: F101»: 60 1’" A 2:! : [00 + x1. [5 it 1 +50 2': I}, c x} : 0’0 + 964 ‘
D V‘st (so = 2 5'
E 15. :1' '16 ‘f' 879
I: X, +l00 : 1" 12. LetU={(m,y,z)eR3jm+2y+z=0}. 2:.) Find a basis of U and give the dimension of U.
b) Find an orthogonal basis of U. c) Find the best approximation to (1, 1, O) by vectors in U. (Wit! ®uzul 5 é('l’*z’§) a ('2, 4’0) ___. 0%.? “é+}(§.3)+I==O .'.b(1€.L‘r/
3 {U‘jdlﬁ 4:) 61M WW M 9% (X. “3+1: '. ”“3."2= it; (H'H'+25)= .32 _ ‘25
"E
on“  ﬁW' .mli ‘01 _
Tm (MW) = ”1' (WIND! + ”'2' (f’if'o) a;
u 11m“ Mm” '
[email protected]+z(o)+(—.L o
‘1‘  v ' _ _ («3) I
Mu(4.i.o)ét( ’ '51(234'0) '+ 553,—: 'E (“(1425)
G) 4 ._ ,L —1
(411°) (2,0;1. = 'EJﬁSL‘JO) “FE? I)Z’S)
: 025%.) e? M40
WWWJgumwnmo ._. g 04— (1 01
(73)/): 2)’2)' 2 —1
13. LetA= —1 2
0 0 GOOD l a) Find the characteristic polynomial det(A — mI) of A, and deduce that the eigenvalues of A
are 1 and 3. b) Find abasis of E1 : {1: E R3  A1: 235:}.
b) Find a basis of E3 = {as E R3  As: = 311:}. d) Find an invertible matrix P such that P“1AP = D is diagonal, and give this diagonal matrix
D. Explain why your choice of P is invertible. a) lA—ztli' ' 13g ”l i: \“53 (9—5:) I 7"! bl l =(3K) (KlvlwiLP l)
3w 12 14. Let u = (1, G, 1) and deﬁne a. linear transformation 6' : R3 —r R3 by S(U)=U,><v, 'UERa, Where “X” denotes the cross product.
a: “*9
a) Showthat S( y )= 56—2
3 y
b) Find the standard matrix of S.
c) Find a. basis for im 5' and describe it geometrically. (1) Find dim ker S. A a"?
i r... ”9  . #3»
X 3 2 ‘6 Amy inns. m0, [MS (1; Leyla»: MmgL. Ol‘wd'u
WMNLcLor (”MI") xC{,o,4) : \f» :o{ :(l 170 H ) =(f.0:4) 01) obt‘mimh'umms mummis Am tors: 3 Ass“: 14 —l. 15. State whether the following are true or false. If true, explain why, if false, give an explicit
counterexample to illustrate. a) A basis of a. ﬁnitedimensional vector space is a. Spanning set which has the largest number
of elements possible. {Lt/ﬂu: ﬁt 1'5 M V... @7— JW go’ﬂ’); (0 I) (44)} Wﬂzwmmfom. gmcx. aﬁtmﬂuz/MJ
Mm+kwmlmfﬁ a} Maid! M
MC}— (mf‘A MT / M 3790/14an 341147 art/4, Mica/cog My W¢W_ 11, $500503”); (9‘1);(OI?}/" .. ’(f”)5 b) Every diagonalizable n x n matrix is invertible. 464224 QM "‘1‘9'4 [Inf] 24 digip ﬁ/Wo! M02,
aaaWJQ/Mn) M a; 441+ meesu) a4 ”Wk/42.141, 16 150) If v E R“ satisﬁes 1)  w for all 10 E R”, then u = 0. t
{I
if
E.
t
{3‘
i
e.
p
0 15d) Let T : R4 —> R6 be a linear transformation such that kerT = {0}. If {'01, v2, 123} is linearly
independent in R4, then {T(v1), T(vg), T(v3)} is linearly independent in R6. We: M. W a7'(142+ .emtmcm;=o ____._——— Mn (Santa "7’41 “UMcm.) 7"{6H4r bw1+cv32
M55) RVE—Eéyhﬁ (it/3’ @kWT':{oSI 94) ...
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 Winter '10
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