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
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
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
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
" 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
(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
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 + :
Math 311-102 (Summer 2006)
Name
1
Test I
Instructions: Show all work in your bluebook. Cell phones, laptops, calculators that do linear algebra or calculus, and other such devices are not
allowed.
1. (10 pts.) Find both the parametric equation for the pla
Math 311-505 (Fall 2003)
Name
1
Test I
Instructions: Show all work in your bluebook. Calculators that do linear algebra or calculus are not allowed.
1. Define the following:
(a) (5 pts.) C[a, b], and its operations of addition and scalar multiplication.
(
Math 311-502 (Fall 2004)
Name
1
Test I
Instructions: Show all work in your bluebook. Calculators that do linear
algebra or calculus are not allowed.
1. (10 pts.) Find both the parametric equation for the plane passing
through the three points P (1, 0, 1),
MATH 304-502/502/506: FINAL REVIEW PROBLEMS
Problem 1
Consider the matrix
1
3
17
5
A = 10
15
2
7
19
(a) Find the LU decomposition of A.
(b) Use (a) to find the determinant det(A) of A.
(c) Find the eigenvalues of U .
Solution.
Part (a). The idea is to app
MATH 311-504: TEST 2 REVIEW PROBLEMS
Problem 1
Let
1
2
A=
1
2
1 1
1 2
.
0 1
0 2
3
2
5
10
(a) Find a basis for the null space N (A) of A.
(b) Let
1
3
b=
2 ,
4
and observe that b can be written as the following linear combination of the columns of A:
MATH 311-504: TEST 1 REVIEW PROBLEMS
Problem 1
Solve Ax = b for
2
1
A=
1
2
3
1
1
2
1
1
1
2
4
1
2
3
9
3
,
5
8
17
6
b=
8 .
4
Solution: Our strategy is the usual one:
(1) Apply Gauian elimination to the augmented matrix (A | b) to get its row echelon form
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
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
Continuous Random Variables
The probability that a continuous random variable, X, has a value between
a and b is computed by integrating its
probability density function (p.d.f.) over
the interval [a, b]:
P(a X b) =
Z
b
fX (x)dx.
a
A p.d.f. must integrate
Discrete Random Variables
Let X be a discrete random variable with
outcomes, x1, x2, ., xn. The probability
that the outcome of experiment X is xi
is P(X = xi) or pX (xi):
i pX (xi) 0
ni=1 pX (xi) = 1
pX is termed the probability mass function.
Joint Di
Marginal Probability Distribution
To compute pX (k), we sum pXY (i, j) over
pairs of i and j where i = k:
pX (k) =
pXY (k, j)
cfw_i, j | i=k
=
j=
pXY (k, j).
Sum of Discrete r.v.s
Let Z be a discrete random variable equal
to the sum of the discrete random
21-325 (Fall 2008): Homework 6 (TWO sides)
Due by October 29, in class
Show FULL JUSTIFICATION for all your answers.
1. Let X be a positive continuous random variable having density fX . Find a formula for the
density of Y = 1/(1 + X).
Solution. To comput
Math 170A
Summer 2010
Quiz 4
Solutions
1. Let X and Y be random variables whose joint PMF is uniform over the set of integers
x and y satisfying 0 x 5 and x y x + 2. Find the following four quantities:
the marginal PMF of Y , E[X], E[Y ], and P(X = 4|Y =