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

Truth Theories
Three Types
Correspondence
Coherence
Pragmatic
Correspondence Theory
A statement is true if it corresponds to or
reflects reality
Reality is something that exists.
Example: rain
Coherence Theory
A statement is true if it is consisten

Gaining Knowledge
Personal Knowledge
Knowledge can be viewed as the production
of one or more human beings. It can be the
work of a single individual arrived at as a
result of a number of factors including the
ways of knowing. Such individual knowledge is

ONE
The Multiple Truths of the Mappable World
Two People, Two Feet Apart
What is the truth? It seems so simple. But when we
try to put it into words, it turns out to be much more
complex.
Our dictionary says that truth is: Conformity to
knowledge, fact, a

The nature of knowledge
Information is acquired by being
told, whereas knowledge can be
acquired by thinking.
Fritz Machlup, 1902-83
Knowledge as justified true belief
Truth
Is independent of.
Belief
Geocentric
model
Heliocentric
model
How do we know some

I know my best friend.
I know how to tie my shoelaces.
I know that 1+1=2.
I know that I dont like eye ball soup.
I know how my friend felt when she broke up with her
boyfriend.
I know that Armstrong landed on the moon.
I know that vert means green in Engl

Ways of Knowing
Emotion
Emotions
Many have felt the power of emotions in
shaping thoughts and influencing behaviour.
There are those who believe that emotions
are an obstacle in the pursuit of knowledge.
Emotion
From Latin movere to move feelings,
pass

Key Terms
For TOK Essay and Presentation
Epistemology
the study of knowledge: how we come to
know things and what the limits of our
knowledge are.
Knowledge
Justified true belief Plato
Validity
Poor justification weak knowledge
Strong justification
Strong

Examiner preparation notes
May 2015
Theory of knowledge
16 pages
2
This markscheme is confidential and for the exclusive use
of examiners in this examination session.
It is the property of the International Baccalaureate and
must not be reproduced or dist

Classnotes - MA2030
Linear Algebra
Arindama Singh
Department of Mathematics
Indian Institute of Technology Madras
This is a modied version of the classnotes of A. V. Jayanthan and A. Singh
Contents
1
2
3
4
5
Vector Spaces
1.1 What is a vector space?
1.2 S

Department of Mathematics, IIT Madras
Quiz-2-Solution ,
MA2030
Linear Algebra & Numerical Analysis
Time: 12/10/2015/8:00-8:50
Max Marks: 20
Answer all questions.
(1) Dene addition and scalar multiplication on R2 by
(a, b) + (c, d) = (a + c, b + d),
(a, b)

Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment - 3
1. Find a basis of V = cfw_(x1 , . . . , x5 ) R5 : x1 + x3 x5 = 0 and x2 x4 = 0.
2. Prove that cfw_1, 1 + t, 1 + t + t2 is a basis for P2 .
3. Determine which

Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Assignment-4
1. In each of the following determine whether T : R2 R2 is a linear transformation:
(a) T (, ) = (1, )
(b) T (, ) = (, 2 )
(c) T (, ) = (sin , 0)
(d) T (, ) = (|

Department of Mathematics, IIT Madras
MA2030
Linear Algebra & Numerical Analysis
Quiz-1 Solution
Duration: 50 minutes
Date: 07.09.2015
Max Marks: 20
Answer all questions. Do not use determinants in any of the calculations.
1. Let A, B Rmn . Show that A is

Next: Definition and Basic Properties Up: Linear Algebra Previous: Similarity of Matrices
Contents
Inner Product Spaces
We had learned that given vectors and
(which are at an angle of
) in a plane, any
vector in the plane is a linear combination of the ve

Rank & Nullity
Denition: Let T : V W be a linear transformation.
Nullity of T = (T ) = dim N(T ).
Rank of T = (T ) = dim R(T ).
Theorem: Let V , W be vector spaces. Let cfw_v1 , . . . , vn be a
basis fof V . Let T : V W be a linear transformation. Then
1

Structure Preserving Maps
1. What are the important maps in real variables?
2. In R the important sets are the intervals.
3. So, what are the maps that take intervals to intervals?
4. In vector spaces, what are the maps that take subspaces
to subspaces?
5

A Short Course on
LINEAR ALGEBRA
and its
APPLICATIONS
M.Thamban Nair
Department of Mathematics
Indian Institute of Technology Madras
Contents
1 Vector Spaces
1.1 Motivation . . . . . . . . . . . . . . .
1.2 Denition and Some Basic Properties .
1.3 Example

Chapter 2
Finite-Dimensional
Vector Spaces
In the last chapter we learned about vector spaces. Linear algebra
focuses not on arbitrary vector spaces, but on nite-dimensional vector
spaces, which we introduce in this chapter. Here we will deal with the
key

Math 100B
Winter 2006
Homework #1
Will Garner
A04528276
Pg 2
#2: (a) Is the set of all positive integers a field? (In familiar systems, such as the integers,
we shall almost always use the ordinary operations of addition and multiplication.
On the rare oc

Math 100B
Winter 2006
Homework #4
Will Garner
A04528276
Pg 32
#1: Suppose that x, y, u and v are vectors in 4; let and be the subspaces of 4
spanned by cfw_x, y and cfw_u, v respectively. In which of the following cases is it true
that 4 = ?
(a) x = (1, 1

Math 100B
Winter 2006
Homework #2
Will Garner
A04528276
Pg 6
#1: Prove that if x and y are vectors and if a is a scalar, then the following relations hold.
(a) 0 + x = x.
This follows directly from A1.
(b) -0 = 0.
Use (c), with a = -1.
(c) a0 = 0.
0 = a

Math 100B
Winter 2006
Homework #3
Will Garner
A04528276
Pg 16
#1: (a) What is the dimension of the set of all complex numbers considered as a real
vector space? (See 3, (9).)
The dimension of will be 2, since it is isomorphic to 2 (over the field ).
(b)

Math 100B
Winter 2006
Homework #5
Will Garner
A04528276
Pg 86
#2: (a) Prove that similarity of linear transformations on a vector space is an equivalence
relation (that is, it is reflexive, symmetric, and transitive).
To show similarity is an equivalence

Math 100B
Winter 2006
Homework #6
Will Garner
A04528276
Pg 123
#1: Given four complex numbers a, b, g, d, try to define an inner product in 2 by writing
( x, y ) = 11 + 21 + 1 2 + 2 2
whenever x = (x1, x2) and y = (h1, h2). Under what conditions on a, b,

Next: Matrix of a linear Up: Linear Transformations Previous: Linear Transformations
Contents
Definitions and Basic Properties
Throughout this chapter, the scalar field
is either always the set
DEFINITION 4.1.1 (Linear Transformation) Let
map
and
or alway