LNMIIT, Jaipur
Department of Mathematics
Programme:
Course Title:
Course Code:
B. Tech. (All
Programmes)
Type of Course:
Linear Algebra and Complex Analysis
MT 121
Prerequisites:
Total Contact Hours:
Institute Core
Calculus and Ordinary Differential Equations
40L+12T
Year/Semester:
Lecture Hrs/Week:
Tutorial Hrs/Week:
Practical Hrs/Week:
Credits:
I/Even
3
1
0
4
Learning Objective:.
This course covers basic notions of linear algebra together with complex analysis. Both the topics have
numerous applications in the areas of science and engineering. The fundamental concept of vector spaces,
linear transformations with their application to the system of linear equations, eigenvalues, eigenvectors
and inner product spaces are to be studied thoroughly in the first half of the course. The second half will be
devoted to discuss analyticity and contour integrals of complex valued functions.
Course outcomes (COs):
On completion of this course, the students will have the ability to:
CO1 Discuss the concepts of vector spaces and Linear Transformations with their
applications to systems of linear equations.
Bloom’s Level
2
CO2 Determine the eigenvalues and eigenvectors of a matrix and its applications
to diagonalization.
3
CO3 Explore the concept of inner product spaces and deduce orthogonal bases.
3
CO4 Interpret the geometry of the complex plane and understand the analyticity
of complex valued functions.
2
CO5 Explain the fundamental concepts of contour integrals and evaluate
improper real integrals using residue theory.
4
Course Topics
1. Unit – I: Vector Spaces, Linear Transformations and System of Linear
equations
1.1. Vector spaces, sub-spaces, linear dependence and independence, linear
span, basis, dimension
Lecture
Hours
5
COs
LNMIIT, Jaipur
Department of Mathematics
1.2. Linear transformations: definition and properties, Rank-Nullity
Theorem and its applications, Matrix Representations, rank of a matrix
(in terms of dimension of range space)
1.3. Finding rank using Row reduced echelon form, Consistent and
inconsistent of system of linear equations, Gauss Elimination Method
2. Unit-II: Eigenvalue, Eigenvectors and Diagonalization
2.1. Eigenvalues, eigenvectors, characteristic polynomial, Cayley
Hamilton Theorem, Minimal polynomial, similarity of matrices,
Diagonalization, Jordan Canonical form
3. Unit -III: Inner product spaces
3.1. Inner Product, Norm, Angles, Orthogonal and Orthonormal sets,
Orthogonal Bases, Gram-Schmidt Orthogonalization process
4
CO1
2
CO2
5
4
CO3
4. Unit- IV: Introduction to Complex Numbers and Function of Complex
Variables
4.1. Complex Numbers, Polar form, Argument, De Moivre’s theorem,
nth roots,
4.2. Neighbourhood, Domain, Function of complex numbers, Limit,
continuity, Differentiability, analyticity, Cauchy- Riemann Equations,
harmonic functions, conjugate harmonic
4.3. Elementary functions of complex numbers: exponential, sin z, cos z ,
log z
5. Unit- V: Contour Integrals and Residue theory
5.1. Path, Length of Path, Contour integrals, ML Inequality, dependence
of integral upon the endpoints of a path, Cauchy-Goursat Theorem,
Generalized Cauchy Integral formula for simply connected domains
5.2. Power series, the radius of convergence, analyticity of power series,
power series representation of analytic function
5.3. Laurent Series; methods of obtaining Laurent Series
5.4. Classification of isolated singularities, Cauchy Residue Theorem,
Evaluation of some real integrals by using Cauchy’s Residue Theorem.
2
4
CO4
2
4
2
CO5
1
5
Text Books:
1. M.T. Nair and A. Singh, Linear Algebra, Springer, 2018. (Unit I - III)
2. Erwin Kreyszig, Advanced Engineering Mathematics, 9th edition, Wiley publishers. (Unit IV- V)
Reference books:
1. G. Strang, Linear Algebra and Its Applications, Thomson Brooks/Cole, 2007.
2. S. Kumaresan, Linear Algebra, A Geometric Approach, Prentice Hall India, 2008.
3. B. Kolman and D.R. Hill, Elementary Linear Algebra with Applications, 9th edition, Pearson Education,
2008
4. J.W. Brown and R.V. Churchill, Complex Variables and Applications, McGraw Hill.
LNMIIT, Jaipur
Department of Mathematics
5. D.G. Zill and P.D. Shanahan, A first course in Complex Analysis with Applications, Jones and Barlett
publishers, 2003.
Additional Resources:
1. MIT 18.06 Linear Algebra by Prof. Gilbert Strang.
Link: https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/
Evaluation Method
Item
Quiz-1
Quiz -2
Midterm Exam
Final Examination
Weightage (%)
10
10
30
50
COs
CO1, CO2
CO4, CO5
CO1, CO2, CO3
CO1, CO2, CO3, CO4, CO5
CO and PO Correlation Matrix
CO
CO1
CO2
CO3
CO4
CO5
PO1
PO2
PO3
PO4
PO5
PO6
PO7
PO8
PO9
PO10
P011
3
3
3
3
3
3
3
3
3
3
1- Low, 2- Medium, 3- Strong
Prepared By: Dr. Ratan Kr. Giri, Dr. Dishari Choudhuri, Dr. Sudipto Choudhuri, Prof. M.K. Kadalbajoo
PO12