Additional Problems 1
1.(Matlab) Construct 4 3 matrix A = (aij ) whose entries are prime numbers with increasing order. For example, the entries in the first and second row of A are 2, 3, 5 and 7, 11
3.6 ROW SPACE AND COLUMN SPACE
KIAM HEONG KWA
p. 154 Let A be the m n matrix. The row space of A is the subspace Row space
of R1n spanned by the rows of A, while the column space of A is Column space
5.1 THE SCALAR PRODUCTS IN Rn
KIAM HEONG KWA
p. 199 The scalar product in Rn is dened by
Scalar product
n
T
x, y = x y =
xi y i
i=1
for any x = (x1 , x2 , , xn )T , y = (y1 , y2 , , yn )T Rn . Note
th
5.2 ORTHOGONAL SUBSPACES
KIAM HEONG KWA
pp. 214-215 Two subspaces X, Y Rn are said to be orthogonal, denoted X Orthogonal
Y , if x, y = xT y = 0 for all x X and all y Y . If X Y , spaces
then X Y = cf
5.3 LEAST SQUARES PROBLEMS
KIAM HEONG KWA
pp. 223-224 In practice, the coefcient matrix A and the right-hand side b of an
m n system Ax = b are obtained emprically. In consequence, the
system Ax = b i
5.4 INNER PRODUCT SPACES
KIAM HEONG KWA
pp. 232-233 An inner product , : V V R on a linear space V is a
function that assigns to each pair of x, y V a real number x, y
with the the following propertie
7.4 MATRIX NORMS AND CONDITION NUMBERS
KIAM HEONG KWA
p. 237, p. 404 A linear space V is called a normed linear space if there is a func- Normed
tion : V R, called a norm on V , satisfying
space
Nonne
ERRATA IN THE CLASS NOTES
KIAM HEONG KWA
Sec. 2.1, p. 1 Add the following before example 1.
Warning! The MATLAB function det(A) may return inaccurate
results due to round-off error. For instance, the
Homework List for Math 571 (Summer 2011)
Kiam Heong Kwa
July 16, 2011
Remark 0. Unless otherwise stated, the exercises are taken from Steven J. Leon, Linear Algebra with Applications 8/e.
Remark 1. Th
Quiz 1 of Math 571 (Su11)
June 23, 2011
The Ohio State University
This quiz consists of two (2) pages and two (2) problems and is worth a total of 50 points.
The point value of each problem is indicat
Quiz 2 of Math 571 (Su11)
June 30, 2011
The Ohio State University
This quiz consists of two (2) pages and three (3) problems and is worth a total of 50 points.
The point value of each problem is indic
Quiz 3 of Math 571 (Su11)
July 7, 2011
The Ohio State University
This quiz consists of three (3) pages and three (3) problems and is worth a total of 50
points. The point value of each problem is indi
Quiz 4 of Math 571 (Su11)
July 14, 2011
The Ohio State University
This quiz consists of three (3) pages and three (3) problems and is worth a total of 50
points. The point value of each problem is ind
Tentative Schedule for Math 571 (Summer 2011)
Mon
June 20
L 1.1, 1.2, 1.3
27
L 2.1, 2.2
Tue
21
L 1.1, 1.2, 1.3
28
L 3.1, 3.2
Wed
22
H&Z 1.1, 1.2, 3.1, 3.2,
3.3
MO 1, 2
29
L 3.1, 3.2
The Ohio State Uni
PARCC Calculator Policy for
Calculator Sections of the Mathematics Assessments
Originally Released July 2012, Updated December 20151
Allowable Calculators
Grades 3-5: No calculators allowed, except f
Multiple Choice Questions
Chapter 16
1. All of the following were anti-Hapsburg leaders during the Thirty Years War EXCEPT
A. Gustavus Adolphus of Sweden
B. Christian IV of Denmark
C. Oliver Cromwell
Age of Absolutism Quiz
Review
CHAPTER 4 SECTIONS 1 & 2
What number is
II?
What number is IX?
What number is XIV?
What is an absolute
monarch?
What was divine
right?
When was the Age of
Absolutism?
Wha
3.5 CHANGE OF BASIS
KIAM HEONG KWA
p. 150 Let E = cfw_v 1 , v 2 , , v n be an ordered basis for a nite dimensional linear space V . The coordinate vector of an element v
V with respect to the ordere
3.4 BASIS AND DIMENSION
KIAM HEONG KWA
p. 138 A nite set of elements B = cfw_v 1 , v 2 , , v n of a linear space V is
said to form a basis of V if
(1) S is a linearly independent set and
(2) V = Span
Additional Problems
1.(Matlab) Create a random matrix A using the command A = floor(10 rand(4, 4). (a) Swap the first and second row of A to get the matrix B using the commands: B = A; B([1, 2], :) =
Additional Problems
1. Consider a set of polynomial, 1 { , ax + b }. 2 (a) Find a > 0 and b such that the above set is an orthonormal set in C[-2, 2]. (b) Find the least squares approximation to ex on
Practice Problems for Midterm 1
8. Consider the invertible matrix 0 1 0 A = 1 1 0 . 4 0 1 (a) Express A as a product of elementary matrices. (b) Using the result of the part (a), compute det(A).
Solutions of Selected Problems in Chapter Test B
3. (a) Note that N (A) = {x R5 |Ax = 0}. We can derive RREF of the corresponding augmented matrix (A|0) as
1 0 0 0
3 0 0 0
0 1 0 0
2 1 0 0
3
Solutions of Selected Problems in HW4
3.1.12 Let x, y be arbitrary elements of R+ . Also let , be arbitrary real numbers. C1: x y = x y R+ , since multiplication of two positive numbers are also p
Solutions of Selected Problems in HW5
3.2.10 (b) { (1, 0, 0), (0, 1, 1), (1, 0, 1), (1, 2, 3) } We have to check existence of C1 , , C4 such that 1 0 1 1 x1 C 1 0 + C 2 1 + C 3 0 + C 4 2 =
Solutions of Selected Problems in HW6
3.4.8 (a) For arbitrary (a, b, c)T R3 , set 1 3 a C1 1 + C2 -1 = b . 1 4 c The augmented matrix is
1 3 a 1 -1 b 1 4 c of which RREF is
Solutions of Selected Problems in HW7
5.2.4 Let y = (y1 , y2 , y3 , y4 )T S . Then < x1 , y >=< x1 , y >= 0 or
1 0 -2 1 0 1 3 -2
y1 y2 y3 y4
=
0 0
.
Solving this equation gives the
Solutions of Selected Problems in HW8
3.5.7 In order to solve this problem, let us recall the following facts; if { u1 , , un } is an orthonormal basis for V and x = c1 u1 + + cn un , then ci =<
Practice Problems for Midterm 1
1. Consider the following linear system
x -y +z = 1 4x +y +z = 5 . 2x +3y -z = c (a) Give the augmented matrix of the system and transform it to a reduced row ech
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