(a) Find an LU-factorization of A i.e. use row operations to find U, an upper triangular matrix equivalent to A and L, a lower triangular matrix such that A LU . (b) Find the determinant of A. 3 1 3 1 4 2 0 and b 1 . 2. Let A 2 2 1 4

(a) Find the determinant of A. (b) Solve the linear system Ax b by the Cramer’s rule. a 3. Let V be the set of all 2 1 real matrices v , where a and b are integers such b 3 8 1 1 that a b is even. Examples of matrices in V are , , , and . 5 2 7 1 Let the operation be standard addition of matrices and the operation be standard scalar multiplication of matrices on V. Is V a vector space? Justify your answer.

4. The following set together with the given operations is not a vector space. List the properties in the definition of a vector space that fail to hold. a V is the set of all 2 1 real matrices v , with operation be standard matrix b addition and the operation be scalar multiplication

c

a c ( a b) b c(a b) , for any real number c.

Note: This assignment must be submitted to your respective tutor (or deposit your assignment in the prescribed MAT111 pigeon-hole on the ground floor of Building G31) on or before Monday, 1 April 2013. Late submission will not be entertained.

...First Semester MATH 1111 LinearAlgebra Chapter 1: Matrices and Systems of Equations
Coverage of Chapter 1: Skip Application 3 in Section 1.4. Skip ‘Triangular Factorisation’ in Section 1.5.
A.
Solving Equations
1.
We are all familiar with solving equations. Illustrate how the following equations can be solved, and then raise and answer some theoretical and/or practical questions concerning the process of solution. (a)
3x 1 8
2 x x 1 (b) 1 2 3 x1 x2 4
(c) (d)
x 4 5 x 2 36 0
x2 x4
B.
Systems of Linear Equations
(Ref: Sections 1.1 and 1.2)
2.
Your first experience of solving systems of equations was probably to deal with a system of two equations in two unknowns. Can such equations always be solved? Are there any special cases?
1
3.
The system mentioned in the previous question is usually called a 2 2 system. How about n n systems in general? (Note: An m n system is a system consisting of m __________ and n __________.)
4.
Solve the following systems of equations. (a)
x1 2 x2 x3 3 x2 x3 2 2 x3 8 x1 2 x2 x3 3 (b) 3 x1 x2 3 x3 1 2 x1 3 x2 x3 4
5.
Look at the previous question again and answer the following questions. (a) Which system is easier to solve? Why? (b) We say that the two systems are equivalent. Explain. (c) Identify three operations in the process of solving for the...

...
KENYA METHODIST UNIVERSITY
END OF 3RD TRIMESTER 2012 (EVENING) EXAMINATIONS
FACULTY:SCIENCE AND TECHNOLOGY
DEPARTMENT:PURE AND APPLIED SCIENCES
UNIT CODE: MATH 110
UNIT TITLE:LINEAR ALGEBRA 1
TIME:2 hours
Instructions:
Answer question one and any other two questions.
Question One (30 marks)
Find the determinant of the following matrices.
-4 8 (2 marks)
0 1
1 -3 -2 (3 marks)
2 -4 -3
-3 6 +8
Find the values of x and y if:(5 marks)
x + 2y 14 = 4 14
-3 y-2 -3 7+3x
Solve the following simultaneous equations using matrix method.
3x + y = 4
4x + 3y = 7(5 marks)
Find the value of K which makes a singular matrix.(3 marks)
3 1
4 -2
4 K 0
Calculate the cross product of the vector U = 2i – 3j – k and V = i + 4j – 2k.(3 marks)
Given the matrices.
2 5 3 -2 0
A = -3 1 and B = 1 -1 4
4 2 5 5 5
Compute:
ATB(3 marks)
tr (AB)(1 mark)
(e) Determine if (2, -1) is in the set generated by = (3, 1), (2, 2) (5 marks)
Question Two (20 marks)
Let T: R2 R2 be defined by T(x, y) = (x + y, x). Show that T is a linear transformation.(7 marks)
Find the basis and dimension of the row space of the matrix.(6 marks)
2 -1 3
A= 1 1 5
-1 2 2
Compute A-1 using row reduction method.(7 marks)
1 4 3
A= -1 -2 0
2...

...Chapter 4 Linear Transformations
In this chapter, we introduce the general concept of linear transformation from a vector space into a vector space. But, we mainly focus on linear transformations from to .
§1 Definition and Examples
New words and phrases
Mapping 映射
Linear transformation 线性变换
Linear operator 线性算子
Dilation 扩张
Contraction 收缩
Projection 投影
Reflection 反射
Counterclockwise direction 反时针方向
Clockwise direction 顺时针方向
Image 像
Kernel 核
1.1 Definition
★Definition A mapping (映射) L: VW is a rule that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set.
★Definition A mapping L from a vector space V into a vector space W is said to be a linear transformation （线性变换）if
(1)
for all and for all scalars and .
(1) is equivalent to
(2) for any
and
(3) for any and scalar .
Notation: A mapping L from a vector space V into a vector space W is denoted
L: VW
When W and V are the same vector space, we will refer to a linear transformation L: VV as a linear operator on V. Thus a linear operator is a linear transformation that maps a vector space V into itself.
1.2 Linear Operators on
1. Dilations(扩张) and Contractions
Let L be the operator defined by
L(x)=kx
then this is a...

...Computer LinearAlgebra-Individual Assignment
Topic: Image Sharpening and softening (blurring and deblurring).
Nowadays, technology has become very important in the society and so does image processing. People may not realize that they use this application everyday in the real life to makes life easier in many areas, such as business, medical, science, law enforcement. Image processing is an application where signal information of an image is analyzed and manipulated to transform it to a different stage. This technique can be done simply by changing the nature of the image using change of basis.
In most situations, people prefer a better image with high resolution, sharper, more detail, etc. Image can be describes as a collection of pixels that have different component depends on the digital signals that digitized as a matrix. These signals came from different energy such as wavelength, frequency. Fourier basis manipulate the image by changing the signal in the pixels. Some signals that give a similar coefficient can be eliminated so that the picture become blurrier or vice versa. These kind functions are found in many situations such as the speeding camera. Speeding camera capture high-speed object, which in return give a result of, blur image. It is almost impossible for human eye to see or track the plate number of the fast moving vehicle without deblurring the image because the range is too high. Fourier change basis is the easiest...

...
Algebra 2 PRACTICE Chapter 12 Test ____________________________ “…………………………..”
3/18/14
You may use a calculator for the entire test; however, the solutions for numbers 1 through 3 must be exact solutions—NO DECIMAL SOLUTIONS FOR THE FIRST PAGE. Do not rationalize. SHOW WORK !
I. Solve the following systems by either the substitution or the elimination (addition) method.
Write your answers as ordered pairs/ordered triples.(These are worth 5 points each)
2. 6x+y-z=-22x+5y-z=2x+2y+z=5
For #3, Solve the system using Cramer’s rule and Algebra. SHOW THE DETERMINANTS and give your solution as a simplified ORDERED PAIR. (exact solutions only). These are worth 4 points each.
Cramer’s ruleSubstitution and/or Elimination
II. Solve the following system by graphing, using the grid provided. (5 points)
324675549466500
-99060-698500 III. Part III tests your ability to use the calculator. You should solve the system by the methods indicated, evaluate the determinants, find the inverse, or perform the matrix arithmetic. Some may be easier to work without the calculator. Give exact answers; give no decimal approximations! If a particular operation is not possible or is undefined, then state the appropriate reason—no work is necessary.
For #5, solve the system using row reduced echelon form or matrix algebra and the inverse matrix. Show the...

...MTH 102
LinearAlgebra
Lecture 12
Projections
Problem
Given a vector 1. 2.
a and a vector b, find p p ∈ span({a}) (b − p) ⊥ a b
such that
a
p =xa
Projections
Problem
Given a vector 1. 2.
a and a vector b, find p p ∈ span({a}) (b − p) ⊥ a b
such that
a · (b − p) = 0 aT b x= T a a
a
p =xa
Projections
Problem
Given a vector
a and a vector b, find p such that 1. p ∈ span({a}) aT b p=a T 2. (b − p) ⊥ a a a a · (b − p) = 0 b a aT b x= T a a p =xa
Projections
Problem
Given a vector 1. 2.
a and a vector b, find p such that p ∈ span({a}) aT b p=a T (b − p) ⊥ a a a Orthogonal b projection of b onto p ∈ span({a}). a p =xa
Projections
Problem
Given a vector
a and a vector b, find p such that 1. p ∈ span({a}) aT b p=a T 2. (b − p) ⊥ a a a Orthogonal What matrix P b projection of b onto p ∈ span({a}). projects b a onto p ∈ span({a})? aaT P= T a a p =xa
Projections
Problem
Given a vector
a and a vector b, find p such that 1. p ∈ span({a}) aT b p=a T 2. (b − p) ⊥ a a a Orthogonal Projection matrix b projection of b onto p ∈ span({a}). that projects vectors a onto p ∈ span({a}). p =xa
aaT P= T a a
Projections
Problem
Given a vector 1. 2.
a and a vector b, find p such that p ∈ span({a}) aT b p=a T (b − p) ⊥ a a a
Orthogonal projection of b onto p ∈ span({a}).
Projection matrix that projects vectors onto p ∈ span({a}).
aa P= T a a
T
Remark: Remark:
P=P P2 = P . Check !
T . Check !
Projections
Problem
Given vectors
a1, a2 and...