(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.

...
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...

...1.1.2
Solving a System of Linear Equations
The system of linear equations (1.3) may be written concisely as
n
aij xj = bi ,
i = 1, 2, . . . , m.
j=1
The matrix (which is a form used in mathematics to denote a rectangular array of
numbers)
⎤
⎡
a11 a12 · · · a1n
⎢ a21 a22 · · · a2n ⎥
⎥
⎢
⎢ .
. ⎥
.
. ⎦
.
⎣ .
.
.
.
am1 am2 · · · amn
is called the coeﬃcient matrix
⎡
a11
⎢ a21
⎢
⎢ .
⎣ .
.
am1
of the system and the matrix
⎤...

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

...MA2030
589
UNIVERSITY OF MORA TUW A
Faculty of Engineering Department of Mathematics B. Sc. 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...

...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...

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