Partial Solution Set, Leon 3.6
3.6.1b We want bases for the row space, the column space, and the nullspace of A =
3 1
3
4
3 1 3
4
1 2 1 2 . Elimination transforms A to U = 0 7 0 2 . We have
3 8
4
2
001
0
one free variable and three nonzero pivots. A basi
Exam 3, make-up
Math 311.501, 503
1. (15 pts.) Dene an inner product on R2 by
$1 91
=2 .
<(I2)(y2)> Im+3xm
2
1
Find
/(Ia) the cosine of angle between ( 1 ) and ( ), using this inner product.
4) the vector projection of ( i ) onto ( _E ), using this inner
" Nam Eff-m 6514 ID 055 jf- 1'11 2
Math 31 I Exam 1 ' Spring 2002
Section 503 P. Yasskin
1. (10 points) AmatrixA satises EgEzEIA = U where
010 100 100 25*
E1: 100 E2= 0313-0 E3: 010 U: 04*
kg 001 001 021 001
and the *s represent unknown non-zero num
Math 3 1 1-503. Fall 06 Name:
Second Midterm
Instructions: Show all ofyoar work. Answers without staiicientjastication will receive little or no credit. The exam contains
a few extra points.
1. (10 points) Exhibit a mattix with eigenvalues 1,0 and eigenve
v E I Ll
N. m 26m m a w 1
Mth311 E '2 S I 2002 1' '
a xam prmg 52 Oasis I3 :15;
E
I
. \
Section 503 P. Yasskin
1. (10 points) Which one ofthe following is NOTavcctor space? Why? g g 7_ 5 R
a. Q: cfw_(*M.v.1c,y,z)ER4 |w+2x+3y+4z=0
- i cfw_lE
MM
EXAM 2, Math 311, Fall 2005
(all problems are worth the same number of points)
Problem 1. Let P1 be the space of polynomials of _ degree at most 1. Let
L : P1 > P1 be a linear- operator whose matrix relative to the basis 1+3, 1 .E
15
12 2
3 4 ' [4
Wri
gages
Exam 2 Math 311-501, Fall 2005
Show your work. No work = no credit.
1. (25pts) The numbers 1 and 2 are eigenvalues of the matrix
5 6 0
A: 3 ~4 0.
1 2 2
If possible, determine an invertible matrix U and a diagonal matrix A such that A =
UAU1 (you do
Dn A^"LI.u,uL
Math 3Ll practice final exam
Instructions: This is a closed-book exam, and no calculators or other electronic devices are allowed. You have 2 hours. Ask for scratch paper if you need it; if you attach more pages, write your
name on every ext
Math 311-503. Fa1106 ' be Name: 23m
_ 7(a) J 8945"!*
Instructions: Show a ofyour work. Answer: without sucientjusncarion will receive lime or no credit. Vectors are denoted
in bold type.
I. cfw_20 points) Let x) be the linear function given by
102 Q1+R13R
Math 311 - 200, Fall 2005
Exam #2 100 points (14 points per problem + 2)
Instructor P. Kuohment
SHOW WORK
GOOD LUCK!
Students name 1_mg_
1. Solve the following systems. Answer the questions:
0 Does the system have a. solution?
0 If it does, is solution
Dir. AmsUcw-c/L
/_._.
Math 311 practice nal exam
Instructions: This is a closed-book exam, and no ealcrators or other electronic devices are al-
lowed. You have 2 hours. Ask for scratch paper if you need it; if you attach more pages, write your
name on e
Exam 3, make-up
Math 311.501, 503
Solutions
. (15 pts.) Dene an inner product on R2 by
$1 :91
1 = 2 +3 .
< $2 > ( 92 )> W mm
a the cosine of angle between 1 and 2 , using this inner
1 1
(
(GMii)
= WHQHIK; >u
x/Ex/Tl J5?
where we have to use the given inne
Name: Tc ALL. r; a 7?: re of F
Math 311 midterm exam I
September 25, 2009
Instructions: This is a closed-book exam, and no calculators or other electronic devices are al-
1owed. You have 50 minutes. Ask for scratch paper if you need it; if you attach
g1,
* 7
/ MATH 311 2006a, Test 1 PRINT NAME: 5 +ep11gns'e E raj-1360? 3
697-13"
1. (10 points). Do the lines 3: =( 1,1,2) +t(2, 4 ,1) and x (0, 3 ,1) +s(1, 2 ,1) intersect? If ARE-Ark.
they do, nd the point of intersection
(I, &)+i(&,4,27=w,%, :)+scfw_-J,
NAME: *7 '31-34
EXAM 1, Math 311, Fall 2005
(all problems are worth the same number of points)
Problem 1. Solve the system 32:) C. [190k 7
11+23322:L'3=1 |-245-'14I~'9
MN I 2213:1423 anew-J
7l 79" ' q
"I: I13 V q
I '1 -'2
. I 7 -2 l
.- I c. |
O] 5/7) _ 0
Named/Inf? PP\ VJ/F
M311 Exam 1 a
Professor E. Howell
2/27/09
Do all problems. The numbers in'parentheses indicate the point value
of each problem. Show all work; answers with no work will not be giVen
(b) (5) How many so 11 mm does Ax : 0 have?
9mg (
Exam 1, version A
Math 311.503
2/16/11
1. (10 pts.) Suppose that A is a four by four matrix and that
a1+a2a3+2a4=0,
where 31, - - -, 34 are the columns of A.
:31 0
(a) Find a non-trivial solution to the system A :2 = 3
x4 '0
(b) Is A nonsingular? Explain
Math 311 200, Fall 2005
Exam #1 50 points (5 points per problem)
Instructor P. Kuchment
SHOW WORK
GOOD LUCK!
f.
Students name _J.J_ 3pr
1. Let a = (1., ,1, 0, 3),b = (2, 4,1,0) 6
o What vector space do these vectors belong to?
- Find 2a 3b
0 Find "all
I F
cfw_.1chva raver-N
Exam 2, version A
Math 311.503
3/23/11
1. (10 pts.) Is the set of vectors cfw_ ( i i ) , ( _f _i ) , ( ; 5 ) in R2 linearly independent? Explain
your answer. '
2. (10 pts.) What is the transition matrix from the ordered basis E = [1 + :
(330
MATH 311 2006a, Test 1 PRINT NAME: 6 U445 H a w;
1. (10 points). Do the lines x = (1,1,2) +(2,4,1) and x 2 (0,3,1) + s(1,2, 1) intersect? If
they do, nd the point of intersection. .
("iMWFN (H
2+ 4 : 81? 1(304? Saar? Run1:5
. +3
iqb 371-8 3 S33: F'f
t . Math 311-503. Fall 06 @ Name: W
Second Midterm E \ Qt WMM
Instructions: Show all of your work. Answers without suicient justication will receive little or no credit. file exam contains
afew extra points.
.l.(10 points) Exhibit a matrix with eigenvalue
52 4'0?
0"
MATH 311 2006a, Test 2 PRINT NAMEJ lame U me man i
2 11 I
1. (10 points). Let f[x)=1 3 U x. Find a basis for the nuIISpace of f.
A? 0
2 2. (10 points). Let f(x) = (
\wch a(
I
7.
1
M 3. (10 points). Let G : OWN00,00) + OWN00,00) be the linear fu
tl
-J6,^l.-cfw_rt^o^
Name:
n
6-a,sr-
Math 311 midterm examI
September25,2009
l*cfw_*tt'
I
I
2
t
L)
A
K
? 0? 02A20 tr
Total
ef
-< , h
Instructions: This is a closed-book exam, and no calculators or other electronic devices are allowed. You have 50 minutes.
bra-WM germ '053-7-fiuall W Ci?
Name_._._~- 1D_._._
Math 311 Exam 3 Spring 2002 i a
' 1? 1 . I
Section 503 P. Yasskin 5 '1 56 9 i 5 24
' 1 3 W20 10EC| 7-?- lo1
Multiple Choice (3 points each.)
Circle 3 to grade for part credit: 1 2 3 <9 (9 6 G)
1. I
journey of Waterloo and Quatre Bras, a journey
which thousands of his countrymen were then
taking. He took the Sergeant with him in his
carriage, and went through both fields under
his guidance. He saw the point of the road
where the regiment marched into
dinner. He sipped Madeira: built castles in the
air: thought himself a fine fellow: felt himself
much more in love with Jane than he had been
any time these seven years, during which their
liaison had lasted without the slightest
impatience on Pitts parta
noon of next day. James Crawley, when his aunt
had last beheld him, was a gawky lad, at that
uncomfortable age when the voice varies
between an unearthly treble and a
preternatural bass; when the face not
uncommonly blooms out with appearances for
which R
Math 311-503. Fall 06
First
Name:
Midterm
Instructions: Show all ofyonr work. Anmers withont stiie
in bold type.
I. [ 20 points) Let x) be the iineormetion given by
1 0
f(x)=(0 1
1 1
ientjitstieation wiil receive little or no credit. Vectors are denoted
)
/
Btlou)ufq, ToA)erl4N
Name:
Math 3lL midterm examII
October30.2009
1
r
2
3
tr
A
+
L,
( o7 9 V I
Total
j9
x2
fnstructions: This is a closed-book exarn, and no calculators or other electronic devices are allowed. You have 50 minutes. Ask for scratchpaper i