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1. For a nonhomogeneous system of 5 equations in 14 unknowns, answer the following three
questions: 1 L4
0 Can the system be inconsistent? / W4 5— ! ]
0 Can the system have a unique solution? w .
0 Can the system have inﬁnitely many solutions? rap” A. No, Yes, Yes.
B. Yes, Yes, No.
Yes, No, Yes. . No, No, Yes.
E. Yes, Yes, Yes.
F. No, Yes, No. 3 5 s
'1 2 —1 1 2, 1 «1+
2 Compute the determinant I”: _01 ﬁ _ ‘1?) 7—+
1 0 3 2 l 6} a '7.
A.6 I
B.—6
I@l2 09'er 3 g 3 I I I
D. 12 :: 1 o t 3 1
E. 18 3 I L! 0 ,.. I
F. 18
l 5 "L o . 2 I 7'47 3. The vectors 1&1 (1,2,4),u2 (2,—3, 1), and 1:3 (2, 1,—1) are orthogonal. Find the
Fourier coefﬁcients of o 2 (3, 5, 2). That is, ﬁnd (a1, a2, 0.3) such that 'v = (1111.1 + 62152 + (13113. a4: ((,1,<p)'(3,$;1)__ 3+{0+8= f (+CF—f(é cu
a2 _ (4,*?,4)»(2,52) : game: ,5
4+?+r :4 M
1 l 2
4. Suppose A = l 0 1 Which one of the following statements is true '?
2 1 4
ILA—1 doesnotexist. '54 1 l I O 0
The third row of A“ is (—1, 1, 1).
C.ThesecondrowofA‘1is(1,2,—l). l O I O 1 o
D. Theﬁrstnowofzél'1 is (2,0,—1). ‘9“ 1 [4 O O f
U
E. The second column of 44*1 is 2 .
1 N [ o p o r 0
F.EachofB,C,D,Eistrue. o I 1 l l 0
O I 7. 0*2 I
r” S {u
4 0 I o f 0 [00 9? F“?
m 0 I I (F.( o (v 010 *2 0 “"1
' oo 1 ‘{ ——( f
o O l 441 _ 5. 6. Which of the following are bases of R3 ?: l a l f O ‘l {9
(1) {(1! 0: I), (0: 4! l): (—4: _4s I 0 q I {(11 1: 1)1(3: _1'.I 2)! (Us 4! (0 Ll N ~ 0 (l' 19 3)! (35 1: _3)! (4: 2! f q ‘~{ cl 0 ‘44 0 0 [1
A. (2) and (3) : z
@831???) ‘ (a) hwy/«5 a( (R?
. 0111; (3) o
E. (1) and (2) H
F. None of these sets is a basis of R3. (1) t l A I t
3 ﬂ 1 "V O we —I
o Lg l 0 q i If A is a 6 X 9 matrix, then the column vectors of A cl A. span a space of dimension 6
B. span a. space of dimension 9 eHess
C. are always linearly independent I I,
D. are sometimes linearly dependent/'5 3mm ska/w M 0} 7 of mm
are always linearly dependent ‘/
. span a Space of dimension 54 at" m”
7. What is the dimension of the subspace of R4 spanned by (1, —1, 4, —5), (2, 1, 5, —1), (0, 1, —I, 3) and([’'[/lno)? g“? (vaf~§ 7{I 4'”;  Elf _
0 13;” 0 3 “3 6’
F‘5 I O 1*! 3 “/ 4 0
/ 0 0 0 5: ~1—l<{~f
” 223$? WWWw” m 8. Suppose V is vector space and that the set {'01, . . . '07} of nonzero vectors spans V. Which fth fll ' t t ? m.————
o e o owmgs a ements are TRUE I“ Adm V2]
(1) dimV=7 mr WW3 P _ ‘ pat/Mng (II) Any 8 vectors in V are linearly [email protected] 7:
(III) V has a spanning set with 8 vectomﬂ ((9. {an I .. , 0;, 1),} (IV) V has a linearly inde endent set with 6 vectors
(V)Igdimvg7 @ ‘9me dr'xm V26, A. I and II. um. dag P B. II and III.
II, III and v.
. II and III.
E. II, IV and v.
F. I, III and V. 4M‘ 18 —7 352203] Merv/8+3 / = [ 8 —3, there is an invertible matrix P such that P—1A2003P is: 31v9~5é+9v
r8 4% c at») ’2 = (9 HM» 4) i
i
[
[ﬁg] A uw%a¢.+2r4
[
[ :beﬂP: :Z O
[a 43 M7 M W @WW .. [2515,30 Qua30....
a 1:
 O 601“ _ o *1 10. Which of the following formulae deﬁne a. linear transformatiou T : R2 —+ R3? (1) T($,y)=($—y.ya$+1) X 7620) 1? T (0.0%)) 0. TON)
(11) They) = ($,y,w+y) / (210“) ‘7’: = 69/0, 0/
(111) They) = (wyway) X _ A.Iand II. Wort): (0,44) 3‘ CL” A, Mi 3113:5531 W = (W) @333 70,4) = (4 Mat T((r°’+T(0:4) £72) (dam
.111 only. MAE 1“ WWW 5? *I If;
(I j '1 11. Suppose p and q E R, A = 3
Z a.) Find rankA and rank[A I b] for all values of p and q.
f b) Find all p so that the column Space of A, col (A), is R3.
.. a” c) Find all p and q so that b g; col (A) ' 3 01) Find all p and (1 so that the system A3 = b has inﬁnitely many solutions. I S—e) In case (d) above, give a. complete geometric description of the set of solutions. ’a1~(’31
{ lot *8 [012 [or
L N *6 N Oil
2 319 “I 0 3133 T6 0 0P“; was {i E; :ii § 0’
MEA@_ (7 31 PM o—rqsoEGD 4») ciIAVﬂE — f; 6
c) tw£ﬁﬂea [H5] 41" mover/ed 6:) MWanJA
<53) f=édz7£~0 ' . .‘:__‘_:_I.;_‘....__.ﬁ .I ....._:f.:;.. I.; _,._ _.. __... .L ._:‘__._*_._._ _ __:_.._.:.:‘.'.w..«_ __.5;_ 12. LetU={(a:,y,z)eR3 x+y——z=0}.. {, f 8.) Find a. basis of U and give the dimension of U.
/ , S‘b) Find on orthogonal basis of U. L5" 0) Find the best approximation to (0, 1,0) by vectors in U. 1' f d) Extend your basis of U in (b) to an orthogonal basis of R3. ' gl‘ma (tiff!) J (fD a) {$73ng; 4) “a mag nud S‘m mm! Mo: 3 Pom a! r723. 2 —10'
13. LetA: —1 2 0. 0 0 3 / a.) Show that 1 and 3 are the eigenvalues of A.
I b) Find a basis of E1 = {:12 e R3  A3 = x}. 2 c) FmdabasisofEa={meR3lAm=3m}. g..— D. Show why your choice of P is invertible. 7 (1) Find an invertible matrix P such that P‘lAP = D is diagonal, and give this diagonal matrix a» w a a: 12> 1—2;!
I 0 a '5'} {cw/K141“? .2 (54‘) N‘WW‘F”)
li'if’w : 6..» (AMArs)
w\_ wpvmp Ashg _
a) L; :WKQJM [pr/:20] 9.. 1.12%] a, [ twp] 3:: Q
’1 O o 2‘) 00° 0 ho
.xpfﬂmo ('MJJ“) war/ng 5/ @
.._. A i J,
c3951 M42?) m: w —_; In: :3] :
o 000‘. OaOdht
2(‘lx4tw/(O’ofngqmd‘312%)150* Mil—E5
.1 H __ ’[ __. *4 ‘
(UM if [Hg] 39““ (PM)— [ogeo] .
001 005 3
19‘ Axum/man. q I}!
14. Let 1: = (1,0, 1) and w = (1,1, 1), and deﬁne a. linear transformation S: R3 —+ R3 by S(u)=ux(vxw), ueR3. / a) Show that if u = (3,9325) 6 R3, then 3(a) = S(:n,y,z) = (y, —m — z,y).
/ b) Find the standard matrix of S. 1c) Find a basis of kerS and give a. complete geometric description of ker S.
/ (1) Find the dimension of im S. / 3) Show that {11, w} for a basis of im S.
2 J' k 3 £0) : 40; : .401)
4)4ﬁx¢0 /{({/ (2.”): (I! @
1‘
( ~410—!
o l D
_ A M <9
mm: awn ~ [2 ‘: c1531,» {(—qu (é
. 090°  d1 ‘ :aa‘waoé“): WA=Z(M(¢U.@
A) arm—{M 04 mg.) w qu+v ,Maa we MES, MAWum, {ﬂoong
M fﬁchhLchw‘é ‘5’”“Wf'h‘9/ fog; (saw wwid w(‘/), 13 .1, Wu..” ~
rt/W"'0l I” “W ' —— l!
15. State whether the followinger true or false. You must explain your answers. (a) If U and V are different 1dimensional subspaces of R2, then their union
UUV={wER2lwerrw€V} isalsoasubspaceofRz. WC“; («3. (A = [XGQ‘I (b) Vectors u, v and w are linearly independent if u is not a multiple of v or 11). MP: [’3' {6:640}; d)‘:0=w' \ W “L Ame/Aw; Wm M {aimleé MW. (c) If A15 3. 4 by 3 matrix and the row canonical form of A has a. row of zeros, then A3: = O has inﬁnitely mag solutlons. ﬂ PALS: l .
J ' ~
15 I (cont)   d) IfAisa12><12 matrix, and A2=—A, then detA=Oor detA=1. Rug
1— \“Lch 2f (51%, on} (ELMOM 3—0 (MA) :9)
WM "L MA at m gamma4) avo  (was. 1‘) If :r : R2 —t R3 is a. linear transformation, then dim(im T) 2 2. FA“; (Q'Wg‘m 7+ dim, W7— ? dirt/t4 //2.2 :91)
: MMT“; £5. 72477) = (0,020) t at
‘ ﬂaw/L c'm T «2: 0. 16 ...
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This note was uploaded on 09/28/2011 for the course MATH 1341 taught by Professor Daigle during the Winter '10 term at University of Ottawa.
 Winter '10
 DAIGLE

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