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, 13, respectively. Write down the Matlab command
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(S ).
pp. 139-140 Let V be a linear space. If there is
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 ordered basis E is the unique vector c =
(c1 , c2 , , cn )T R
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
the subspace of Rm spanned by the columns of A. The ran
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
that
x, y = xT y = y T x = y , x .
The Euclidean length o
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 = cfw_0. To see this, let z X Y , so that z 2 =
z , z = 0 a
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 is usually inconsistent. In this case, we look for a
vec
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 properties:
(1) x, x 0 for any x V with equality only if x = 0.
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
Nonnegativity: p 0 for all p V ;
Positivity: p = 0 only if p
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 matrix A =
123
1 0 1
3 2 1 has the row reduced echelon
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. This list will be updated throughout the quarter.
Remark
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 indicated. You may use MATLAB to check your results.
However,
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 indicated. You may use MATLAB to aid your calculations. Howe
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 indicated. You may use MATLAB to aid your
calculations. How
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 indicated. You may use MATLAB to aid your
calculations. Ho
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 University
Thu
23
Quiz 1
H&Z 1.1, 1.2, 3.1, 3.2,
3.3
MO 1,
3.3 LINEAR INDEPENDENCE
KIAM HEONG KWA
p. 128 Let V be a linear space and let u, v k V , k = 1, 2, , n. Then
the statements
(1) u Span(v 1 , v 2 , , v n );
(2) Span(u, v 1 , v 2 , , v n ) = Span(v 1 , v 2 , , v n ).
are equivalent.
n
Proof of (1)(2). Stat
3.1/3.2 DEFINITONS AND EXAMPLES OF VECTOR SPACES
AND SUBSPACES
KIAM HEONG KWA
p. 113 A (real) linear space or a (real) vector space is an ordered triple
(V, , ), where V is a nonempty set and : V V V and
: R V V are functions, such that the following axio
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], :) = B([2, 1], :) What relation between det(A) and det
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 the interval [-2, 2] by a linear function (a firs
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).
Solution.
(a) Three consecutive row operations on
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 positive. C2: x = x R+ , since exponential funct
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 = x2 0 1 1 3 x3 for arbitrary x1 , x2 ,
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
1 0 4a - 3c -a + c 0 1 . 0 0 -5a + b + 4c
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 =< x, ui >. Moreover, by the Parseval's equality, th
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 echelon form. (b) Decide how many solutions the syste
1.1 SYSTENS OF LINEAR EQUATIONS
KIAM HEONG KWA
pp. 1-5 A linear system of m equations in n unknowns or an m n
linear system is a system of the form
(I)
a11 x1 + a12 x2 + + a1n xn = b1 ,
a21 x1 + a22 x2 + + a2n xn = b2 ,
.
.
.
am1 x1 + am2 x2 + + amn xn =
1.2 ROW ECHELON FORM
KIAM HEONG KWA
p. 13 A matrix is said to be in row echelon form if
(1) the rst nonzero entry in each nonzero row is 1;
(2) if the k th row does not consist entirely of zeros, the number of
leading zeros1 in the (k + 1)th row is greate