1. Assignment 1 (PART B)
Submission date: 6 October 2015, Time 3:pm
Question 1(CLO 1): Suppose that the weather in a certain city is either rainy or dry. As
a result of extensive record keeping it has been determined that the probability of a rainy
day fo

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Recall
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Recall
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Recall
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Why Linear Independence
Each vector in the vector space can be built up from the elements
in this generating set using only the operations of addition and
scalar multiplication. The generating se

Department of Electrical Engineering
Instructor:_
Date: _
Course/Section:_
Semester: _
EE221: Digital Logic Design
Lab1: Familiarization of Basic Gates and Digital ICs
Name
EE221: Digital Logic Design
Regn. No.
Demo/Report
Viva
Total
Marks / 25
Marks /10

PAM 242: LINEAR ALGEBRA
Credit Hours: 2
Instructors /Course Coordinators Name:
Muhammad Shahid Nazir
Textbook:
Lay D. C. Linear Algebra and its Applications, Pearson Education, 4 th Edition, 2012.
References and other Supplemental Material:
1. A. Howard,

Linear Algebra
Eigenvalues, Eigenvectors
Eigenvalues and Eigenvectors
If A is an n x n matrix and x is a vector in
R n , then
there is usually no general geometric relationship
between the vector x and the vector Ax. However,
there are often certain nonz

Simultaneous Linear Equations
Topic: Gaussian Elimination
12/04/15
http:/numericalmethods.eng.usf.e
du
1
Gaussian Elimination
One of the most popular techniques for
solving simultaneous linear equations of the
form A X C
Consists of 2 steps
1. Forward Eli

Linear Algebra
Linear Transformations
Real Valued Functions
Formula
Example
f x
f x x
f x, y
Function from R to
R
2
2
f x, y x y
f x, y , z x 2
f x, y , z
f x1 , x2 , , xn
Description
f
2
x1 , x2 , , xn x1
Function from
2 to R
R
2
Function from
y2 z2

linearly independent
The set of vectors v1 , v2 , . . . , vm in a vector space V are called
linearly independent if the equation
c1 v1 + c2 v2 + . . . + cm vm = 0
(2)
holds if and only if
c1 = c2 = . . . = cm = 0
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linearly dependent
The set of vect

Example Show that the set of vectors (1,3,-1),(2,1,0), (4,2,1) span
R3 .
Solution:
(a, b, c) = c1 (1, 3, 1) + c2 (2, 1, 0) + c3 (4, 2, 1)
(c1 + 2c2 + 4c3 , 3c1 + c2 + 2c3 , c1 + c3 ) = (a, b, c)
Two vectors are equal if their corresponding components are

Main Contents
Matrices
Linear System of Equations and their solutions
Vector Spaces
Eigenvalue problems
Linear transformations
Inner product spaces
Least Square Curves
1 / 34
How does linear algebra help with computer science
graph analysis
3D transformat

Matrix Multiplication
Multiplication of a Matrix by a Matrix
The product C = AB (in this order) of an m n matrix A = [ajk ]
times an r p matrix B = [bjk ] is dened if and only if r = n and
is then the m p matrix C = [cjk ] with entries
n
cjk =
ajl blk = a

Trace of a square matrix
The trace of a square matrix A is the sum of the diagonal elements.
Properties:
tr (A + B) = tr (A) + tr (B)
tr (cA) = ctr (A)
tr (A) = tr (AT )
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Invertible or nonsingular matrices
A square matrix A is said to be invertible

Example: Obtain a matrix in
equivalent to
0
0
A=
2
2
row echelon form that is row
2 3 4 1
0 2
3 4
2 5 2 4
0 6 9 7
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1
Find the rst (counting from left to right) column in A not all
of whose entries are zero. This column is called the pivotal
col

normal matrices
A real matrix A is normal if it commutes with its transpose, i.e.,
AAT = AT A
Examples of normal matrices: symmetric, skew symmetric and
orthogonal matrices.
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normal matrices
A real matrix A is normal if it commutes with its transpos

LU factorization
An LU-factorization of a square matrix A is of the form A = LU,
where L and U, for example for a 3 matrix is given by
1
0 0
u11 u12 u13
1 0 , U = 0 u22 u23
L = m21
0
0 u33
m31 m32 1
A linear system is given as Ax = b, we can write it as

Example: Determine the current that ow through each segment
of the circuit.
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Solution:
Step 1 By Kirchhos Current Law (KCL)
inow outow
nodP i1 + i3
i2
nodQ
i2
i1 + i3
So from above information, at nod P we have i1 + i3 = i2 . At nod
Q, we have i2 =

Example: Determine the current that ow through each segment
of the circuit.
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Markov process
Application 4 Note: A stochastic matrix is also called transition,
probability or markov matrix.
Theorem
If T is the stochastic or transition matrix of Marc

Vector space
Let V be an arbitrary nonempty set on which two operations are dened:
addition, and scalars multiplication. We call V a real vector space if the
following axioms are satised.
1 If u and v are objects in V, then u + v is in V
a u + v = v + u,

Linear combination
Let v1 , v2 , . . . , vn be vectors in a vector space V . The vector v in V
is called a linear combination of v1 , v2 , . . . , vn if v can be written as
n
v = c1 v1 + c2 v2 + . . . + cn vn =
ci vi
(1)
i=1
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Example Show that wheth

Linear Algebra
Systems of Linear Equations
and Matrices
Linear Equation
Alinear equationis anequationin which
eachtermis either a constant or the
product of a constant times the first power
of a variable. Such an equation is
equivalent to equating a
first