Welcome to MTH 102 !!
Linear Algebra
&
Ordinary Differential Equations
Contact Information
Instructor-in-charge
Office
Phone
Email
Web
: Prawal Sinha
: 516, Faculty Building
: 7213
: prawal@iitk.ac.in
: www.home.iitk.ac.in/~prawal
Co-Instructor
Office
Phone
Email
Web
: Akash Anand
: 574, Faculty Building
: 7880
: akasha@iitk.ac.in
: www.home.iitk.ac.in/~akasha
Agenda
Administrative stuff !
Introduction to Linear Algebra
System of Linear Equations
Geometry of Linear Equations
Method of Solution
Evaluation Policy
•  Quizzes
50
•  Mid Semester exam
60
•  End Semester Exam
90
-----------200
Course Information
•  Course webpage for Linear Algebra :
–  http://home.iitk.ac.in/~akasha/mth102
•  Lecture slides will be available on the course webpage.
•  Homework :
–  Problem sets will be made available on the course webpage
and at Copy Point in the shopping center.
–  Weekly Problem Sets will be available by Friday evening of
the preceding week.
•  Quizzes :
–  There will be two quizzes, one covering Linear Algebra and
the other covering Ordinary Differential Equations.
–  Tentative dates are February 2 and April 6, 2013. If there are
any changes, you will be informed in advance.
Reference Material
•  Introduction to Linear Algebra
–  Gilbert Strang
•  A First Course in Linear Algebra
–  Robert A. Beezer
(Available at http://linear.ups.edu/download.html)
•  Professor P. Shunmugaraj’s Lecture Notes
(Available at http://home.iitk.ac.in/~psraj/mth102/
lecture_notes.html)
Agenda
Administrative stuff !
Introduction to Linear Algebra
System of Linear Equations
Geometry of Linear Equations
Method of Solution
Introduction to Linear Algebra
•  Solve n linear equations in m unknowns
a11 x1 + a12 x2 + · · · + a1m xm
=
b1
a21 x1 + a22 x2 + · · · + a2m xm
..
.
=
b2
..
.
an1 x1 + an2 x2 + · · · + anm xm
=
bn
Introduction to Linear Algebra
•  Electrical networks
i1
i1 − i2 − i3 = 0
10i1 + 10i2 = 10
10i1 + 20i3 = 5
A
3×3
10 Ω
i2
10 V
A
10 Ω
i3
system of linear equations !
20 Ω
5V
B
Introduction to Linear Algebra
•  Electrical networks
… requires solving a system with a large number of linear
equations and unknowns …
Introduction to Linear Algebra
•  Differential Equations
d2 x(t)
m
= −kx(t)
2
dt
Introduction to Linear Algebra
•  Differential Equations
mx¨1 (t)
=
mx¨2 (t)
=
�
−2kx1 (t) + kx2 (t)
kx1 (t) − 2kx2 (t)
�
�
k 2
x¨1 (t)
=−
x¨2 (t)
m −1
−1
2
��
x1 (t)
x2 (t)
�
… solution method requires computing “eigen values”, etc. … Linear Algebra !!! …
(we will learn how in the second half of the course!)
Introduction to Linear Algebra
•  Differential Equations



2
x¨1 (t)
k
x¨2 (t) = − −1
m
x¨3 (t)
0
−1
2
−1
… large systems …


0
x1 (t)
−1 x2 (t)
2
x3 (t)
Applications of Linear Algebra
•  Applications in almost all scientific and
engineering disciplines !
•  For example,
–  Image Compression
–  Cryptography
–  Genetics
–  Graphs and Networks
–  Computer Graphics
–  …
Agenda
Administrative stuff !
Introduction to Linear Algebra
System of Linear Equations
Geometry of Linear Equations
Method of Solution
System of Linear Equations
•  Solve n linear equations in m unknowns
a11 x1 + a12 x2 + · · · + a1m xm
=
b1
a21 x1 + a22 x2 + · · · + a2m xm
..
.
=
b2
..
.
an1 x1 + an2 x2 + · · · + anm xm
=
bn
System of Linear Equations
•  A particular interesting case : n linear
equations, n unknowns
a11 x1 + a12 x2 + · · · + a1n xn
=
b1
a21 x1 + a22 x2 + · · · + a2n xn
..
.
=
b2
..
.
an1 x1 + an2 x2 + · · · + ann xn
=
bn
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
•  Matrix Form
�
�� � � �
−3 x1
0
=
1
x2
2
1
−1
A
x
b
Ax = b
System of Linear Equations
Definition: Matrix
An m × n real matrix is a rectangular layout of
numbers from R having m rows and n columns.
An m × n complex matrix is a rectangular array of
numbers from C having m rows and n columns.

1
2
A=
3
4
2
3
4
5

1
1

1
1
A 2×3
complex matrix.
A 4×3
real matrix.
�
1+i
2
2
3i
3
4 + 5i
�
System of Linear Equations
Definition: Matrix
An m × n real matrix is a rectangular layout of
numbers from R having m rows and n columns.
An m × n complex matrix is a rectangular array of
numbers from C having m rows and n columns.
Notation
For a matrix A , the
entry in row i, and
column j will be
denoted by [A]ij .
System of Linear Equations
Definition: Matrix
An m × n real matrix is a rectangular layout of
numbers from R having m rows and n columns.
An m × n complex matrix is a rectangular array of
numbers from C having m rows and n columns.
�
� � � A column
� �
Convention
1 −3 x1 vector0 of
� An m × 1 matrix
� �−1
is � 1 � x� =
size 2 .
2
1 −3 x
0
also known as a
1
COLUMN VECTOR
2
of size m .
−1
1
x
=
2
System of Linear Equations
Definition: Matrix
An m × n real matrix is a rectangular layout of
numbers from R having m rows and n columns.
An m × n complex matrix is a rectangular array of
numbers from C having m rows and n columns.
�
2 3 4
�
A �row vector �
of size
2 3. 4
Convention
A 1 × n matrix is
also known as a
ROW VECTOR of
size n .
System of Linear Equations
Matrix Operations: Addition




aij
A





m×n

+

bij
B





m×n

=

cij
C
cij = aij + bij
Symbolically,
[C]ij = [A + B]ij = [A]ij + [B]ij




m×n
System of Linear Equations
Matrix
Operations:
Scalar
Multiplication






α


aij
A
Symbolically,





m×n


=


bij = αaij
αaij
B
[B]ij = [αA]ij = α[A]ij





m×n
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
1
−1
�� � � �
−3 x1
0
=
1
x2
2
•  Matrix-Vector Multiplication
�
1
−1
�� � �
�
−3 x1
x1 − 3x2
=
1
x2
−x1 + x2
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
1
−1
�� � � �
−3 x1
0
=
1
x2
2
•  Matrix-Vector Multiplication
�
a11
a21
a12
a22
��
�
�
x1
a11 x1 + a12 x2
=
x2
a21 x1 + a22 x2
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
1
−1
�� � � �
−3 x1
0
=
1
x2
2
•  Matrix-Vector Multiplication
�
a11
a21
a12
a22
��
�
�
�
�
x1
a11
a12
= x1
+ x2
x2
a21
a22
linear combination of columns
�
System of Linear Equations
•  General linear system

a11 x1 + · · ·
+
a1n xn = b1
a21 x1 + · · ·
+
..
.
a2n xn = b2
am1 x1 + · · ·
+
amn xn = bm
a11
 a21

 ..
 .
am1
···
···
..
.
···



b1
a1n  
x
1
 b2 
a2n 
  ..   
..   .  =  .. 
 . 
. 
xn
bm
amn
•  Matrix-Vector Multiplication




a11 x1 + · · · + a1n xn
a11 · · · a1n  
 a21 x1 + · · · + a2n xn 
 a21 · · · a2n  x1

  ..  

=
 ..




.
.
.
.
..
.. 
..
 .


xn
am1 x1 + · · · + amn xn
am1 · · · amn
System of Linear Equations
•  General linear system

a11 x1 + · · ·
+
a1n xn = b1
a21 x1 + · · ·
+
..
.
a2n xn = b2
am1 x1 + · · ·
+
amn xn = bm
a11
 a21

 ..
 .
···
···
..
.


am1
···
•  Matrix-Vector Multiplication

a11
 a21

 ..
 .
am1
···
···
...
···




b1
a1n  
x
1
 b2 
a2n 
  ..   
..   .  =  .. 
 . 
. 
xn
bm
amn


a1n  
a11
a1n
x1
 a21 

 a2n 
a2n   . 




..   ..  = x1  ..  + · · · + xn  .. 
 . 
 . 
. 
xn
amn
am1
amn
System of Linear Equations
•  A 2 by 2 example
�
�
�� � � �
−3 x1
0
=
1
x2
2
1
−1
x
A
1
−1
−3
1
A
0
2
b
�
b
Augmented
Matrix
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
−3
1
0
2
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
Interchange
the equations
−x1 + x2
x1 − 3x2
=
2
=
0
�
�
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
−3
1
0
2
Interchange
rows
-1
1
1
-3
2
0
�
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
Interchange
the equations
−x1 + x2
x1 − 3x2
=
2
=
0
�
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
−3
1
0
2
New system has same solution
as the original system.
�
-1
1
1
-3
2
0
�
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
−3
1
Multiply an equation with
a non-zero constant
x1 − 3x2
−cx1 + cx2
=
0
=
2c
�
0
2
�
Multiply a row
with a constant
1
-c
-3
c
0
2c
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
Multiply an equation with
a non-zero constant
x1 − 3x2
−cx1 + cx2
=
0
=
2c
�
−3
1
0
2
�
New system has same solution
as the original system.
1
-c
-3
c
0
2c
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
Multiply an equation with
a non-zero constant and
to another equation
x1 − 3x2
−x1 + x2 + c(x1 − 3x2 )
−3
1
0
2
�
New system has same solution
as the original system.
=
0
=
2 + c(0)
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
�
Multiply an equation with
a non-zero constant and
to another equation
x1 − 3x2
−x1 + x2 + c(x1 − 3x2 )
Easy to check that x1 = −3
and x2 = −1 solves the
system of linear equations
1
−1
�
−3
1
1
-1+c
=
0
=
2 + c(0)
0
2
-3
1-3c
�
0
2
�
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
Definition:
Equivalent Linear Systems
Two systems of linear equations
are equivalent if their solution
sets are equal.
x1 − 3x2
−x1 + x2 + c(x1 − 3x2 )
=
0
=
2 + c(0)
System of Linear Equations
•  A 2 by 2 example
x1 − 3x2 = 0
−x1 + x2 = 2
Equivalent
Linear Systems
x1 − 3x2
−x1 + x2 + c(x1 − 3x2 )
x1 − 3x2
−2x2
c=1
=
0
=
2 + c(0)
=
0
=
2