This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Page 1 of 7 DO NOT OPEN THIS BOOKLET UNTIL YOUARE INSTRUCTED TO DO SO ECE 2:331  Exam 2 —— Spring 2008 I agree to abide by the provisions of the University of Houston Academic Honesty Policy while taking
this exam. Signature: ‘1 "r r.
Date: H ,_ r i “’3, E... r: INSTRUCTIONS: (1) Read and sign the Academic Honesty statement above. Unsigned exams will not be graded. (2) Including this cover page, this exam has 7 pages. Verify this now and raise your hand if you have a
problem. Do NOT look at the questions!  (3) This exam is closed book and closed notes except for one page(two sides) of notes which must be handed
in with your exam. Be sure your name is on the notes page so it may be returned to you. (4)Communication devices of any kind (eg. cell phone, pager, etc.) may not be used during the exam. Turn
them off or give them to the room monitor. No such devices should be visible on your desk. (5)The use of a calculator is prohibited! All steps must be shown (6)You will have 90 minutes for this exam.
(7)When you finish the exam, unless the twominute warning has been given, put your crib sheet inside the exam, hand in your exam, and leave the room.
(8)0nce the twominute warning is given, stay seated until time is called, at which point you must stop writing immediately and stand up. Continuing to work after time is called will result in substantial penalty. Problem 1. /6 pts Problem 4. /lOpts Problem 7._ /10pts Problem 2 l 0 /lOpts Problem 5. % /l4pts Problem 8. /15pts Problem 3 /10pts Problem 6 l /10pts Problem 9. i S /15pts Sol TOTAL SCORE /100pts possible Page 2 of 7 Problem 1(6 ptstotal)
A. (3 pts) Find the equation of the line through the points (2,4,1) and (6,6,5)
x __ I “m a! m; a: a m /
2 .3“ x was t B.(3 pts) Find the perpendicular distance between the line you found in part A. and the line through the points
(3,0,1) and (5,4,4). [3i *4" m t‘ {u l m '  I . h ,. Problem 2. (10 pts)Use Gerschgorin’s Theorem to ﬁnd the greatest integ lower bound and the smallest
integer upper bound for the real eigenvalues of the matrix ——2 4 —3
g: —1 —1 2
2 0 0 Page 3 of '7 q Problem 3. (10 pts) If a matrix A has the following two ei genvalue/ eigenvalue pairs, ﬁnd the matrix A— 3 —l e * ‘33; m I " 
—0.5, l, a. V3 . g [a ‘2} i} [:2 " . if; . 
1 2 4.... m a ) _ Egr: r ii iii “3% "m ‘1 “V33 “2% ‘3 W” ‘ N ‘,
* i ‘3 3‘ «i 62 J '72 . r f e i i W W iii a “I "i 3 i“ "am . ,. .s ,5, i “L”: 'i
but" 2:: i ii ii i Q l i i Q i if ii W % iii
* D l q a» * a; a. "3* l t a? t F; ﬂ “" .
A!“ l d m 5 ﬂ 3 “l “with m if agg igg‘ I!”
2 P b P I E. I ’9 i éﬂg‘jfi I’J’i a; i“
'a ﬁ‘“ i 4 r: 1 E J a if '
F’ i; j :1; E if 5 it w an "" g f M a i i ii" a :5 :7
if ' w} i1 1 q} H I? g ’L if 15‘ j .}‘/
.w "2 mi; E. ,,;_ ..2. ML {.43 r
($143} “’HT} ("Ohm “’1 6 W3») 1% *ii 7 3f ,. 3:5 .11 l
a in? ~ “5,1 J, '3 Zita a Li .— +1 :”“”*" I!
aw “a 4..” WW ,_ r w
i. MiaE rim 1‘
I 5 g
A "I: .1,
l5 IQ _ HIMFWW . . Magnumru. uh i'ﬂfroliléhﬁi“4.(10pts) Determine wli'either—or—not the following three vectors are linearly independent.
State clearly your conclusion and the reason for it. 2
2
2
2 i": ii ll
#00 '15 II
U.)th Page 4 of 7 Problem 5.(l4pts total) 0 0 0
A (6 pts) Find the adjoint for matrix A = 1 1 1 T
'  l _ a a r i} "1 2 3 “Z” I [1 3l ' l 3} * 1 :9 ’3
i“ ~IH I? 33! w53:! 0 a? «'5 “ r “I? :"l I??? I.
'4'. ' ' I? _ I H
. . II .
1  K in; H _ w ' an? E 2,. "L I;
21 2 is “2;. I ““ [:2 '1‘ [3 r lg. .n __ ' 5'2. {'2' Page 5 of7 Problem 6.( 10 pts total, 2 each)Select the best answer for each of the following and circle your choice,
A. M311: . .ismawsguacemSu‘ix and X2 = [0], then which of the following is NOT necessarily true? Pin. _  ' !_ (G)D_2>det(X)~—— 0
~‘**”"‘°Z (fix: [0] (d) the rows of X are linearly dependent
(c) all of the above must be true B. If A is a 3x5 matrix, B is a 4x3 matrix, and C is a 5x4 matrix, which of the following is NOT deﬁned?
(a) (MTV (e) more than one of the above is NOT defined
C. %he nxn matrix A is invertible, then as) be an 6i <1
E”...
E
o
H":
3;. ._...'r~:.',  .nl: ' A must be d.i.§gQ.11§1hle.:.   m1mr mm 1n .'.;:I..1::¢r'..':<'ﬁ'.:'' 4/2, (o) A cannot have repeated eigenvalues
(d) if A is real it cannot have complex eigenvalues D. If matrix A has 4 rows and 4 columns and det( A )=2.l, then the system of equations ﬁg = i; has: (a) no solution; CE exactly 1 solution;
{HZF (C) exactly 2 solutions; an infinite number of possib solutions; 'W ﬁnﬂmﬁwam' 9"“ “minu. " all dfthé”§5‘5%“§fé possible. ,i IIIIIIII H. “EmulateMin". rrthJrv‘W' PL'l‘ﬁ'T.'.‘1'I'FH:I'W. MWWH‘T'KLLm aw L as or .. E. If matrix A has 4 rows and 4 columns and rank( g )= 4, then the system of equations _A_i I B has: (b) no solution; c exactly 1 solutlon; i. ' (d) exactly '2 So utlons;
(e) an inﬁnite number of possible solutions;
(f) all are possible. Problem 7.( 10 pts)If A z , find the approximation to an eigenvector after two iterations of the power method, beginning with the vector . "can! _ Page 6 of 7 1 3 3 2
Problem 8.( 15 pts total) Consider the system AX=B where A = 2 6 9 5
—1 —'3 3 O
0
(10 pts)A..If B at 0 , what condition(s) must the components of B satisfy in order for AX = B to be
0
consistent?
a an
1 .
(5 pts)B. .If B = 5 , ﬁnd the general form of solutions to AX=B and express your answer as a linear
5
combination of column vectors. I. x
*2; j; Z a; ' ﬁt. 4 1W2; “'* 5*” “i3 "3' *5 it :3 a
C. a 5' w i m a»? ﬁ' M "'13 2’?
z wt my
K1 1, f are “t as
:62” f w '2: ““t
s Page 7 of ’7 2 4 —— 6
Problem 9.(15 pts) The matrix A z — 2 8 — 6 has eigenvalues 3,4,4. Find a maximal linearly
— 1 2 1 independent set of eigenvectors and indicate whether A is diagonable. 3‘3. ii HQ, “I it “tic; {ml Jf é: ‘3 i‘ “a; {1? a
._ I "
"2 3"5 Hg, 1:. h? S “"5 ‘: ‘3’ "1' in? “3 6’ ‘5' 1 a i «’2. if
Wm“ 7‘ ﬂ 3 ; vi 2 am} "I! 1 “a? b _ Lg) *2. ‘i it} ‘3? .31 "i 31% ...
View
Full Document
 Spring '08
 BARR
 Linear Algebra, Linear Independence, Vector Space, pts

Click to edit the document details