3 (16 1313s.) Suppose a, n, In are a basis for a subspace of R4, and those ar-o the columns
of a matrix A.
(a) How do you know that ATy : E] has a solution 3; 75 D?
(b) How do you know that AI = [J has o-nlI the solution :5 = U?
4 (25 13133.) Suppose A is a 4 by 3 matrix, and the complete solution to
1
U D
4
Art": 1s 3: 1 +e1 2
1
1 1
1
(a) What is the third column of A?
(b) What is the second column of A?
[(3) Give all known information about the rst column of A.
2. Suppese- the matrix A is this prcduct BO (net L times HE):
200 2244
A=BC=340 0000
201 2266
(16) (a) Find bases fer the raw space and the cclurnn space cf A.
(8) (1) Find a basis fer the space cf all selutiens tc A2: = U.
(8) (C) All these answers will
3.
(12) (3.) Find the row-reduced echelon fern] R [if A and alse the inverse matrix
E1 that preduees'A = ElR,
1 ['13 3
A: 2 [J 5 5 . FindRandE'l.
1133
(9) (1) Separate that multiplieatien ElR into eeiumns of E1 times rows of R.
This allows you to write A
18.06 Professor Strang Quiz 1 Solutions March 6., 1-998
1. (a) The nullspace has dimension 2'. Therefore 3 T : 2 and r : l.
(b) The rst column- of A comes from knowing the particular solution. The other
columns come from knowing the two special solutions
18.06 Professor Streng Quiz 1 March 6, 1998
Your name is:
Please circle your recitation:
1) M2 2-132 M. Nevins 2-538 3-4110 mouicamsth
2) M3 2-131 A. Voronov 2-224 3-3299 veronovi'rueth
3) T10 2-132 A. Edelreu 2-380 3-7770 edelmeureth
4) T12 2-132 A. Edel
18.06 Professor Strang Quiz 1 March 10, 1999
Your name is: Grading
#WMIl
Please circle your recitation:
1) Mon 23 2-131 S. Kleiman 5) Tues 121 2-131 S. Kleiman
2) Mon 34 2-131 S. Hollander 6) Tues 12 2-131 S. Kleiman
3) Tues 1112 2-132 S. Hovvson 7) Tue
4.
(16) (a) 'Suppcse A is' an m by a matrix cf rank. 1. Describe Exactly the matrix Z
(its shape and all its entries) that ccrnes frcrntranspcsing the rcw- echelon
form of R (prime Ineans transIJcise):
Z = rref(rref(A_)'_)'.
(7) (b) Ccrnpare Z in Prchlern
18.06 Quiz 1 October 8. 1999 Closed Book
Your name is: Grading 1
2
Please circle your recitation: i
1) M 2 2-131 W; Fong 2) hi 2 2-132 L. Nave
3) M 3 2-131 W; Fong 4) T 1U 2-131 H. Matsinger
5) T 19 2-132 P. Clifford 5) T 11 2-131 H. Matzinger
7) T 11 2
[12]
U 3
4. Suppose :r: = _1 is the only solution to A1: = 5
U 7
9
4a. Fill in each (blank) with a number.
The columns of A span a (blank) _din1eusiona1 subspace of the vector space Rmnk).
The columns of A span a "3dimensional subspace of the vector space
1010101010
0101010101
1010101010
0101010101
(3-)11: 1010101010
' 0101010101
1010101010
0101010101
1010101010
0101010101
1 1
2. (20 pts) Sketch the image of the square gure to the left below after applying the map. A = ( ) .
' 1 3
You may use the graph pap
4 (31 pts.) (a) To nd the rst eelnnfn'l nf A'l (3 by 3), what system ALE : I) would
yen salve?
[13) Find the rst column .nf A1 (if it exists) fer
[132
A: 13!]
16:0
(:3) For each a and I), nd the rank sf this matrix A and saI why.
([1) Far each a and i), n
1 (.30 pta.) Suppeee the matrix A has reduced rea:r echelon ferm R:
1212; 1203
A=2a18,R=0012
(rew3) 0000
(:3) What can yu say immediately abcut raw 3 cf A?
(b) What are the numbers a and b?
(c) Describe all aeiatieae .ef Re: : [1. Circle the spaces that a
18.06 Professor Strang Quiz 1 Solutions October 81 1999
3
4
(3) NM) = M3) and C(AT) = C(BT)
1 U 2 7
2 U 1 [l
(b) 0 , 1 for the row space, 0 , _5 for the nullspace.
7 5 CI 1
(c) rIlrue
Reason: 'Whenever a combination EEC-I- dy = U, multiply by A to see tha
3 (25 late.) Suppose A is a 3 X 5 matrix and the Solutions to' ling : D are 'epauue'd by
the veoto re
1 1 [J
y: 1 , U , 1
l] 1 1
(a) What is the rank of this A?
(b) For all A, why does the rank of A equal the rank of the blook matrix
B: ?
AA
(o) If the ra
2 (21 late.) A is m by n. Suppeee Am : b has at cfw_least [me Sam-em fer every I).
(a) The rank of A is
(13] Describe all E'ECtDIE in the nullepeee ef AT.
[:3] The equation ATy : C has (0 er'1)(1 er BOND er DO)(1) ee'lutien fer
every :3.
(b) Fer each e, What is the rank of A?
(c) Fer each c1 describe exactly'the nullepece cf A.
[cfw_1 F01" each c1 give a basi'efer the column Space cf A.
2 (25 1313s.) Suppese
1 0 0 1 0 1 4 5
A: 1 1 0 0 1 2 :2 1
712 00011
(a) Find a basis fer the nullspace of A.
[13'] Find a basis fer the (3011111111 space of A.
(:3) Give the complete solution to
An: = 3
21
18.06 Professor Edelrnan Quiz 1 October 1, 1998
Your name is .
Please circle your recitation:
T12 2132 Anda Degeratn [email protected] 2229 31589
) T1 2 21 31 Edward Goldstein egoldmath 2092 36228
10 T1 21 31 Anda Degeratn [email protected] 2229 31 589
1 1 T2 21 32 Tue L
18.06 Professor Strang Quiz- 1 Solutions March 101 1999
1 (a) 1. (no c)
2. (allci)
3- (320
(b) I'Etllli3 (37$ 9
rank2 cZ'U
(c) N(A) = cfw_D ifc 7E U
_'2
N(-A] : all multiples of 1 if c = D.
0
(d) c 7% 0 Give any basis for R3-
0 1
c : [J one basis is [J ,
3. (3.0 ptsj) Please briey but clearly explain your answers.
(3.) Are the set of invertible 2 "x 2 matrices in M' a-snbspace?
(13.) Are the set of singular 2 i: 2 matrices in M a subspace?
(c.) Consider the matrices in M whose nnllspace contains
1
Is this
Math 544, Final Exam Information.
Wednesday, December 8, 2 - 5, LC 115.
The Final exam is cumulative.
Useful materials.
1. All homework.
2. Exams I, II, III, and solutions.
3. Reviews for exams I, II, and III.
4. Class notes.
New Topic List (not necessari
Math 544, Final Exam. 12/8/10.
Name:
No notes, calculator, or text.
There are 100 points total. Partial credit may be given.
Write your full name in the upper right corner of page 1.
Number the pages in the upper right corner.
Do problem 1 on page 1,
Math 544, Exam 1 Information.
9/14/10, LC 115, 2:00 - 3:15.
Exam 1 will be based on:
Sections 1.1 - 1.3, 1.5 - 1.7, 1.9;
The corresponding assigned homework problems
(see http:/www.math.sc.edu/boylan/SCCourses/544Fa10/544.html)
At minimum, you need to u
Math 544, Exam 1. 9/14/10.
Name:
Read problems carefully. Show all work.
No notes, calculator, or text.
The exam is approximately 15 percent of the total grade.
There are 100 points total. Partial credit may be given.
Further instructions.
Write your
Math 544, Exam 2. 10/10/10.
Name:
Read problems carefully. Show all work.
No notes, calculator, or text.
The exam is approximately 15 percent of the total grade.
There are 100 points total. Partial credit may be given.
Further instructions.
Write you
Math 544, Exam 2 Information.
10/12/10, LC 115, 2:00 - 3:15.
Exam 2 will be based on:
Sections 1.7, 1.9, 3.2, 3.3, 3.4;
The corresponding assigned homework problems
(see http:/www.math.sc.edu/boylan/SCCourses/544Fa10/544.html)
At minimum, you need to un
Math 544, Exam 3 Information.
11/18/10, LC 115, 2:00 - 3:15.
Exam 3 will be based on:
Sections 3.4 - 3.7, 4.2 - 4.4;
The corresponding assigned homework problems
(see http:/www.math.sc.edu/boylan/SCCourses/544Fa10/544.html)
At minimum, you need to under
Math 544, Exam 3. 11/18/10.
Name:
No notes, calculator, or text.
There are 100 points total. Partial credit may be given.
Write your full name in the upper right corner of p.1.
Number the pages in the upper right corner.
Do problem 1 on page 1, probl
4.
(b) The third remr of R is zero! Sc. the two columnrow multiplications are from
I columns 1 ant-12 of E1 and rows 1 and 2 of R:
1 _0 1033 0000
2[1.033]+[1[0100]:2056+000_0
1 1 1033 010-11
(51) The matrix Z is m by a. All its entries are zero except for
18.06 Professor Str-ang Quiz 1 February '26, 1997
1. (36: 4 times-9 points] A is a 3 by 4 matrix and b is a_ column vector in R3:
1322 :2
A=2768 5:?
3967 r
(a) Reduce A-J. : b -to echelon form Us: 2 c and nd one solution I? (if a solution
exists].
(b) Fin
3. A is an m by 11 matrix of rank 1". Suppose As: : E) has no solution Jfor some right sides
It: and innitely many solutions for some other right sides I).
(a) (9) Decide whether the- nullspace of A contains only the zero vector and why.
(h) (9 points] De