# Advanced Linear Algebra

By Prof. Premananda Bera   |   IIT Roorkee
Learners enrolled: 2393
ABOUT THE COURSE:
This course is designed to provide students with an understanding of Mathematical concept on Linear algebra that includes basic as well as advanced level. Attempt is taken to cover both Both theoretical as well as computation perspectives. There are six componenets:
i) Linear System of equations,
(ii) Vector spaces,
(iii) Linear transformations,
(iv) Cannonical forms and Jordan forms,
(v) Inner product spaces and different operators in it,
(vi ) Bilinear and Quadratic forms,Orthogonal projection and Spectral theory, and
(vii) Singular value decomposition.

INTENDED AUDIENCE: Master students of Mathematics,,Physics, B.Tech III Year Electrical, Computer Science,

PREREQUISITES: Linear Algebra B.Sc /BE I/II year
Summary
 Course Status : Completed Course Type : Core Duration : 12 weeks Category : Mathematics Credit Points : 3 Level : Postgraduate Start Date : 23 Jan 2023 End Date : 14 Apr 2023 Enrollment Ends : 06 Feb 2023 Exam Registration Ends : 17 Mar 2023 Exam Date : 30 Apr 2023 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

1.1 System of linear equation (Review)
1.2 Elementary matrix operation, elementary matrices,
1.3 Rank of matrix, Matrix inverse
1.4 Vector spaces,
1.5 Subspaces
1.6 Bases, and dimension
1.7 Ordered basis, coordinate matrix
1.8 Computation concerning subspaces
1.9 Linear transformation,
1.10 Existence of Linear transformation
1.11 Rank Nullity theorem(Review)
1.12 Representation of transformations by matrices
1.13 Change of ordered basis and matrix representation of transformations
1.14 Algebra of Linear Transformation
1.15 Linear operators and Linear Functional
1.16 Dual Space, Double dual spaces
1.17 Transpose of Linear transformation
1.18 Characteristic values, diagonalization, (Review)
1.19 Annihilating polynomials
1.20 Invariant subspace
1.21 Triangulation (2)
1.22 Simultaneous triangulation and simultaneous diagonalization
1.23 Direct sum decomposition
1.24 Invariant direct sums
1.25 The primary decomposition theorem
1.26 Jordan forms
1.27 Rational form
1.28 Inner product spaces
1.29 Gramian matrix, Gram Schmidt orthogonalization
1.30 Orthogonal complements, Best approximation
1.31 Operators on Inner product spaces
1.32 The adjoint of a linear operator
1.33 Normal and self adjoint operator
1.34 Unitary and orthogonal operators and their matrices
1.35 Bilinear and Quadratic forms
1.36 Orthogonal projections and the spectral theorem
1.37 *Generalized g-inverse of a matrix, The Singular value decomposition

### Books and references

(i) Linear Algebra (second edition) Kenneth Hoffman, Ray Kunze (Pearson)
(ii) Advanced Linear Algebra (Second Edition) Stevan Roman (Springer)
(iii) Matrix and Linear Algebra K. B. Datta (Prentice Hall of India)

### Prof. Premananda Bera

IIT Roorkee
Dr. Premananda Bera is a Professor in the Department of Mathematics, IIT Roorkee. Currently, he is also the Head of the Department. He has more than twenty years of teaching experience for both undergraduate and postgraduate courses. He has taught more than 14 applied and pure mathematics courses. His research area is focused on computational fluid dynamics and hydrodynamic stability theory. He has published more than 35 journal articles.

### Course certificate

The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 30 April 2023 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
Registration url: Announcements will be made when the registration form is open for registrations.
The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Roorkee. It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.

- NPTEL team