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 way to transform the image. Eigenvector change basis will probably give the best result of transformation, but the problem is it will cost a lot of time, effort, and also expensive.

The development of image processing applications has many impacts on the society. Television for example, in the early century, television can only transmit and receive monochromatic moving images that give people black and white or grayscales images. This kind of television was not comfortable for human eye because it is not clear, detail...

...
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...

...Engineering Level 2 - Semester 2 Examination: MA 2030 LINEARALGEBRA Time Allowed: 2 hours
2010 September 2010
ADDITIONAL
MATERIAL: None
INSTRUCTIONS
TO CANDIDATES:
This paper contains 6 questions and 5 pages.
Answer FIVE questions and NO MORE. This is a closed book examination.
Only the calculators approved and labeled by the Faculty of Engineering are permitted. This examination accounts for 70% of the module assessment.
Assume reasonable values for any data not given in or with the examination paper. Clearly state such assumptions made on the script.
If you have any doubt as to the interpretation of the wording of a question, make your own decision, but clearly state it on the script.
- 1-
MA2030
1. a) Let a be an object and define V = {a }. On V, define the addition as
a+a = a.
ra
Define the scalar multiplication as
=a
V scalar r .
Prove or disprove whether V becomes a vector space under these operations. (7 marks)
b) For each of the following cases, prove or or disprove whether S is a subspace of V. (7 marks)
i)
V = the vector space of all the n x n matrices. S= { A
IA
V.
E
V and At = A }
ii) V = any inner product space Let
Uo E
s = {u I U E V, < U , Uo > = I}
2. Let V be a vector space. a) Suppose
Prove that
{ Ul ,U2 ,
{ Uj
.u; } are linearly dependent if and only if at least one in
,U2 ,
.u.; } can be expressed as a linear...

...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...

...The purpose of this project is to solve the game of Light’s Out! by using basic knowledge of Linearalgebra including matrix addition, vector spaces, linear combinations, and row reducing to reduced echelon form. |
Lights Out! is an electronic game that was released by Tiger Toys in 1995. It is also now a flash game online. The game consists of a 5x5 grid of lights. When the game stats a set of lights are switched to on randomly or in a pattern. Pressing one light will toggle it and the lights adjacent to it on and off. The goal of the game is to switch all the lights off in as few button presses as possible. In the folling examples, 1 will represent a “on” light and 0 will represent an “off” light. Yellow represents a button pressed and changed and green represents a button that was not pressed but was changed as a result of the pressed button.
Example 1
1 | 1 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 1 |
0 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 1 |
0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
Starting Grid Pressing button 1 Pressing button 18
Example 2
1 | 1 | 0 | 1 | 1 |
1 | 0 | 1 | 0 | 1 |
0 | 1 | 1 | 1 | 0 |
1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 1 | 1 |
0 | 0 | 0 | 1 | 1 |
0 | 0 |...

...Luc Brubaker
LinearAlgebra Essay
Computer Graphics
As technology has advanced, LinearAlgebra has proven to be a very vital part of the graphic imaging process. Computer graphics is the use of computers to produce pictorial images on a video screen, or a computer screen. Graphic software uses matrix mathematics to process linear transformations to render these images. Through 4 x 4 matrices, we are able to project any three-dimensional image onto a two-dimensional screen. Not only are these images projected, but through the manipulation of these matrices and once they are on the screen, we are able to take the two-dimensional and move them as well as scale them to different sizes. Through various techniques, we are able to produce realistic pictures without ever having to take out a camera. This is why linearalgebra opens up the doors for so many possibilities.
Video games are a great example of what linearalgebra can create because they are all virtue and contain zero real life scenes. Square matrices can represent linear transformations of geometric objects. Through several linear transformations, a viewer can feel as if they are experiencing the geometrical objects first hand. For example, in the Cartesian X-Y plane, the matrix with a negative one in the...

...4 2 1 0 4 6 1 3 . 1. Let A 8 16 3 4 20 10 4 3
(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...

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