(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.

...2012–13 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...

...
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...

...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...

...
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....

...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...

...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...