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Unformatted text preview: Page 1 of 7 DO NOT OPEN THIS BOOKLET UNTIL YOUAREINSTRUCTED TO DO SO ECE 2331  Exam 2 — Fall 2003 I agree to abide by the provisions of the University of Houston Academic Honesty Policy while taking
this exam. Signature: —l—.‘ '3 Date: ll— \‘5 “33> 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)00mmunication 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 prohibitedi All steps must be shown (6)You will have 90 minutes for this exam. (7)When you finish the exam, unless the two—minute warning has been given, put your crib sheet inside the
exam, hand in your exam, and leave the room. (8)0nce the two—minute 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. #5 Probleml. SQ /15pts Prob 4. iii/(Q /20pts Prob 6. 3 l15pts
Problem2 3' /15pts Prob 5. 322/0 /20pts Problem3 1—5 /15pts ‘ TOTAL SCORE .7." 3;. /100pts possible GRADE % Page 2 of 7 Problem 1(15 pnts total) 1 1
0
A.IfA= —1 2 and 52E ],ﬁndBZAT
y 2 3 B. Find a unit vector (vector of length 1) that is eﬁhogenal to both u : [—1 ,2,2] and v = [2,1,3] C. What are the parametric equations for the line passing through the points (4,3,1) and (1,},0) ? Page 3 of7 Problem 2. (15 puts total)
Select the best answer to each of the following and circle your choice \ l 2 ~ 4 2
A. Which statement best descnbes the matrix 0 1 — 2 —l .
l 0 l 2 1. rows dependent, columns independent
2. rows dependent, columns dependent 3. rows independent, columns independent
4. rows independent, columns dependent 2 — 3‘ — 4
B. Which statement best describes the following set of vectors: 1 , 1 , — 2
3 —1 — 6 1. linearly independent 2. linearly dependent 3. a basis 4. possibly the column vectors of a system AX=B which has a unique solution
5. more than one of the above is true 1 0 2 0
C. Ifthe reduced row echelon form of A is 0 1 — 3 0 then a basis for the solutions to AX=[O]
0 0 0 1
would be:
1 0 0 _ ' — 2
1 0 0
0 1 0 3 _ . .
l. , , 2. 0 , l , 0 ® 4. There IS no 133515 for the solution set
2 — 3 0 1
0 0 1
0 0 1 0 D. HA is 5x5 and det(A) I 2, then
1. det(AT) = 1/2
2.. det(3A) = 486
3.det(A'l) : 2
4. det(—A) = 2 E. The matrix equation AX = I where A is 4x5 will have a solution for X if
1 rank of A is 4
2. rank of A is 5
3. XA = I is also solvable
4. cannot have a solution since A is not square Page 4 of 7 Problem 3.(15pnts t0tai)Use Gaussian eliminatiOH on the augmented matrix to solve the system below. If
there is no solution, indicate clearly why, if there are an inﬁnite number of solutions, express the general form
as a linear combination of column vectors. 2x1+4x2+x3+7x4 +x5=5
xl+2x2 +3X4K5=2
X1+2X2+2X3+11X4+5X5=4 Page 5 of 7 1
Problem 4.(20pnts)Let A 2 I and B =
2 HNO
WHO
OOH
ONO
4300 3) Find A4 and 3—].
b) Solve the following equation for X: ' (_A_ ting“ —B'])X:BA_1X+ [(A3)_1A2]_1. Page 6 of 7 6 0 2
Problem 5(20 pnts). Let A = 0 2 0 .
—4 0 0 3) Find the eigenvalues and corresponding eigenvectors for A.
b) Find a diagonal matrix D and a nonsingular matrix P such that A = PDP—1 . Note, you do not need to
ﬁnd the matrix PHI. Page 2 of 7 '2‘ M
Problem 105 pnts total) 11111 _1 I, "
1 1 x 0 ‘11 “L %k
A.IfA= —1 2 andB=[2 ],ﬁndBZAT L
2 3 y _
l”). 131 ‘11 1 V6113 ﬁnifﬁ‘, O ‘— 1, ,, V1.11 “a
“(.1379 c: ‘ “ “1L?
t? % 1111111\ 1 31
I“ 111110 3049 7.73%:7 \ ((#11 —:LXL “Lo“ "1
a. ”31“ 1"» "1 “MW" ‘ '“ m u ..'>nt «u 'x H 4
MN m“ a m m War Ws
//
1/ B. Find a unit vector (vector of length 1) that is orthogonal to both 11 : [— 1 2 ,2] and v— _ [ 2,1,3] Mg \___ 5;. ” ‘4'?”"5‘3
\I Hi ‘ i
:> 1'3 1,"'L\\\33 : H1.  7 "
{1:1 ; LU ‘ 1 ’I‘
_ 1; {i \‘H
vwév) 02.6, “‘m‘rk: Pitmj My“) “ PM“) i
' 1f 7 aA 1  ”1
u a: “mama ; =39    _ “ ‘ ‘ ” A M: C. What are the parametric equations for the line passing through the points (4 2 ,1) and (1,1,0) ? ULJ U 'ﬁ'.44‘_1mk‘1 =11". [\w‘vi‘ ~57§1 “’a—icsi r1?
I , Page 3 of? Problem 2. (15 pnts total)
Select the best answer to each of the following and circle your choice \ 1 2 ~4 2 \ 1 —“ 1 z t e a El
0 \ an »\ it) o l —I—;
A. Which statement best describes the matrix 0 1 —2 —l ~> 1 .e D [ lo 0 “ ‘~ l 1 O 1 2 F %‘ _el_t\’?.$'lp\
1  7‘ ’.
1. rows dependent, columns independent l w o 01 gt "g
2. rows dependent, columns dependent k .3 1" q l
 rows independent, columns independent 5:
. ows independent, columns dependent ”
2"—3 ~41: ipw
B. Which statement best describes the following Set of vectors: 1 , 1 , — 2 l i \ , l‘" \L1 .
l g; L: I
3_ —1 — 6 t 3
1. linearly independent 3 «i— 313 :51; cilia. w L i
@linearly dependent ‘3 "W 0 “ “r °” 5 L"‘ ‘
3.abasis 9":“KFVQ'3'Z’Q‘ \ e *1] ..i
4. possibly the column vectors of a system AX=B which has a unique solution X i ‘0 Z "i 5. more than one of the above is true 1 0 2 0
C. If the reduced row echelon form of A is 0 1 — 3 0 then a basis for the solutions to AX=[0] 0 0 O 1 “5‘5””
would be: L“ ,5 5' E1
1 0 0  ”I — 2
1 0 0
0 1 0 3 . . .
1. , , 2. 0 , 1 , 0 ® 4. There is no 133818 for the solution set
2 —3 0 1
0 0 1
0 0 1 0 D. HA is 5x5 and det(A) = 2, then
1. det(AT) = '/2 @rdeteA) = 486
3.det(A“) : 2 gem—A) = 2 E. The matrix equation AX = I where A is 4x5 will have a solution for X if bdrUQ‘: L—‘LC‘l {RT} / @rankofAisdl 2. rank of A is 5
3. XA =1 is also solvable
4. cannot have a solution since A is not square Page 4 of '1' Problem 3.(15pnts t0tal)Use Gaussian elimination on the augmented matrix to solve the system below. If
there is no solution, indicate clearly why, if there are an inﬁnite number of solutions, express the general fonn
as a linear combination of column vectors. 2XJ+4X2+X3+7X4 +X5=5
x1+2xz +3x4—x5=2
x1+2x2+2x3+11x4+5x5=4 1H14l‘3'l \V—D‘aeyl“: F1Log—\L Ft'to‘: ~\''‘]
l 1 D 3—l1—i:\> ". 1“" Ti ' 5 :5 1C3 1 \ i—l—ﬁ‘ o o \ ‘\ ‘5 \
‘1 1. ll '5' L1} \ 'L'l '5 L‘ If: '3 ~=8 G ‘>_, L\ o a "5 G C’l
lamAV; F ‘1“ﬁ"ll’2 lit,—\'§_$w
\ "L o "S "‘l't "‘
o l l 3 l l e e e l b 01 “s
’f: ’Jt.\ " ‘ LlL'L "*lil If 1*: “Lakeelliz {/1/ 11.5: 3 #1 (—1., ° 0 I «Lei 1 I +K‘ '5 +i£1 D
l "Qi D D 0 [Ht 0 t a Pay!“ 0’? Q: :3 Page 5 of”! L? r
I O O 1 0 0 ’5 f 1: _,
‘m' 3» ‘
Prob]em4.(20pnts)LetA: 1 2 1 andB: 0 2 0. , “i: “H“
i: “ 21’ rﬂ " 3:
2 1 3 0 0 4 g 31:17)?
. — —1 “1’ c ” ”3”“
a)F1ndAlandB . ¢_.3,\ ‘_
b) Solve the following equation for X: if 25" T;
—1 ~ W‘s.
—] [_ h
_ _ _ 3 2 L " i” ..:
l’: 3 ‘ (A+B)@ 1"B1)X=BA 1X+[(A ) A : _ is: Li;_.;_
E 2:: g, _ I b (b o 1 ,
.R*::§§mlio%D‘EL1i\ii? globi‘oﬂj “‘603.C’oh‘ r\oo\09
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E (11 “22’5“?‘\ E12 F5453“: V? ‘1F'L .513
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5 r» 6 0 2 Q o m A!
Problem 5(20 pnts). Let A = 0 2 O . D 1 f3; .1 C,
—4 0 0 "'1 e
3) Find the eigenvalues and corresponding eigenvectors for A.
b) Find a diagonal matrix D and a nonsingular matrix P such that A = PDIYl . Note, you do not need to
ﬁnd the matrix P4.
(“'9 MW 2 EM “ii“ If”; 1'91: if‘E—JWIiW‘R'h w}? e um] {01 {a}
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