Linear Algebra and Numerical Analysis
Assignment-1
Jan-May, 2012
In the following V denotes a vector space over F which is R or C, and Fmn denotes the
set of all m n matrices with entries from F.
1. For x, y V , show that x + y = x implies y = 0.
2. Let x
SOLUTION (i) We have the graph of
cfw_
H ( )= 0, < 0
1, 0
SOLUTION (ii) The Graph of
cfw_
H ( )= 0, >0
1, 0
The graph of
cfw_
H (t )= 0, < t
1, t
There are two cases:
Case1:
when t <0
H ( ) H ( t )=0 H ( ) H ( t ) d=0
Case2:
when t 0
The graph of
H ( ) H
SOLUTION:TheFourierintegraloff(x)isgivenby
cos
sin
Where
1
1
cos
sin
cos
Now
1
1
1
cos
4
1
1 sin
4
1 sin
1 sin 2 sin 3
4
1 cos
4 sin 3 sin 2
4
1 5 sin 3 3 sin 2
4
Now
1
1
1
sin
4
1
4
1
4
cos
cos 2 cos 3
sin
1
1
4
4
1 sin
cos
cos 3 cos 2
1
4
3 cos 3 3 cos
SOLUTION5:
SOLUTION7:
SOLUTION10: We have
P=$ 10000, r=2.5 , n=10
r
A=P 1+
100
(
n
)
Since interest compounded quarterly, therefore
P=$ 10000, r=
2.5
, n=10 4=40
4
We get
(
A=10000 1+
2.5 /4
100
40
) =12830.27
The value after 10 years will be $12830.27.
S
QUESTION: A multiple choice test has four possible responses. To test whether this question was answered
correctly by more people than would be expected due to chance, a hypothesis test of 400 student answers is
conducted.
1. What is the null and alternat
9. In using marginal analysis, an additional unit should be stocked if:
A. the probability of selling that unit is less than or euqal to ML/(MP+ML)
B. MP+ML
C. the probability of selling that unit is greater than or equal to MP/(MP+ML)
D. the probability
1
Vector Spaces
1.1
Introduction
The notion of a vector space is an abstraction of the familiar set of vectors in two or three
dimensional Euclidian space. For example, let ~x = (x1 , x2 ) and ~y = (y1 , y2 ) be two vectors in
the plane R2 . Then we have
6
Vector Spaces
1.3
Subspaces
We observe that
V = cfw_x = (x1 , x2 ) R2 : x2 = 0, which is a subset of R2 is a vector space with respect to
the addition and scalar multiplication as in R2 .
V = cfw_x = (x1 , x2 ) R2 : 2x1 + 3x2 = 0 which is a subset of
Basis
1.6
11
Basis
Definition 1.10 (Basis) A subset E of a vector space V is said to be a basis of V if it is
linearly independent and span E = V .
EXAMPLE 1.18 For each j cfw_1, . . . , n, let ej Fn be such that ej (i) = ij , i, j = 1, . . . , n.
Then we
MA 2030
Linear Algebra and Numerical Analysis
Arindama Singh & S. Mishra
Prepared by
A. V. Jayanthan & Arindama Singh
AS & SM 0
Lecture slides, topic wise as we progress, will be available in
http:/mat.iitm.ac.in/home/asingh/public_html/teaching.html
It w
INTRODUCTION:
Counting Principle: When there are
there are
ways of doing both
ways to do one thing, and
ways to do another, then
The fundamental counting principle is a mathematical rule that allows you to find the number of
ways that a combination of eve
QUESTION1:
SOLUTION: (a)
We have
2
y=16 t +0.05 t + 168
Diffrentiating with respect to t
dy
=32 t+ 0.5
dt
When the top of the ladder will reach the ground then velocity will be zero
Therefore,
dy
=0
dt
32 t+ 0.5=0
t=
1
se c
64
SOLUTION: (b)
FIRST Method
B
7
Greens Functions and Nonhomogeneous
Problems
The young theoretical physicists of a generation or two earlier subscribed to the
belief that: If you havent done something important by age 30, you never will.
Obviously, they were unfamiliar with the histor
NOTES ON THE EXISTENCE AND UNIQUENESS THEOREM
FOR FIRST ORDER DIFFERENTIAL EQUATIONS
I. Statement of the theorem.
We consider the initial value problem
cfw_ y(w) = F(w,y($)
y($o) = yo.
(1.1)
Here we assume that F is a function of the two variables (x, y),
Chapter 5
Boundary Value Problems
A boundary value problem for a given differential equation consists of finding a solution of the
given differential equation subject to a given set of boundary conditions. A boundary condition
is a prescription some combi