(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.

...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...

...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....

...LINEARALGEBRA
Paul Dawkins
LinearAlgebra
Table of Contents
Preface............................................................................................................................................. ii Outline............................................................................................................................................ iii Systems of Equations and...

...Solving systems of linear equations
7.1 Introduction
Let a system of linear equations of the following form:
a11 x1
a21 x1
a12 x2
a22 x2
ai1x1 ai 2 x2
am1 x1 am2 x2
a1n xn
a2 n x n
ain xn
amn xn
b1
b2
bi
bm
(7.1)
be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m;
k = 1, 2, ..., n) are the coefficients, bi (i...

...Initialise
Clear the workspace and load LinearAlgebra package
> restart;
> with(LinearAlgebra):
If you want practice at hand calculation you should use the worksheet "Interactive Gaussian Elimination" (see Menu)
Enter the matrix of coefficients and right-hand side vector
You may edit the following statements or use the matrix and vector pallettes to enter new data ( see View, Palettes)
> A:=<<4 | 2 | 3 | 2> , <8 | 3 | -4 | 7> , <4 | -6 | 2 |...

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