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Swayam Central

Linear Algebra

By Prof. Arbind Kumar Lal   |   IIT Kanpur
The course will assume basic knowledge of class XII algebra and a familiarity with calculus. Even though, the course will start with defining matrices and operations associated with it. This will lead to the study of system of linear equations, elementary matrices, invertible matrices, the row-reduced echelon form of a matrix and a few equivalent conditions for a square matrix to be invertible. From here, we will go into the axiomatic definition of vector spaces over real and complex numbers, try to understand linear combination, linear span, linear independence and linear dependence and hopefully understand the basis of a finite dimensional vector space. We will then go into functions from one vector space to another, commonly known as linear transformations. For the finite dimensional case, we will see that all functions can be understood through matrices and vice-versa. We will then define inner/dot product in a vector space. This leads to the understanding of length of a vector and orthogonality between vectors. As our main result in this part, we will understand the Gram-Schmidt orthogonalization process. Finally, we will go into the topic of eigenvalues and eigenvectors associated with a square matrices or linear operators. As a final result, we will learn the spectral theorem for Hermitian/Self-adjoint matrices. As an application, we will classify the quadrics.

INTENDED AUDIENCE
Mathematics Honours
PREREQUISITES : Class XII algebra and calculus
INDUSTRIES  SUPPORT     : None

Learners enrolled: 2040

SUMMARY

Course Status : Upcoming
Course Type : Core
Duration : 12 weeks
Start Date : 20 Jul 2020
End Date : 09 Oct 2020
Exam Date : 17 Oct 2020
Enrollment Ends : 27 Jul 2020
Category :
  • Mathematics
  • Level : Undergraduate/Postgraduate
    This is an AICTE approved FDP course

    COURSE LAYOUT

    Week 1:Introduction to matrices, matrix operations, such as addition of matrices, transpose of a matrix, scalar multiplication and matrix multiplication etc. Invertible matrices, examples and submatrix of a matrix
    Week 2:System of linear equations, elementary row operations, elementary matrices, Gauss elimination and Gauss – Jordan methods, LU decomposition, Row-reduced echelon form of a matrix, Rank of a matrix and the solution set of a linear system
    Week 3: Application of the solution set of a linear system to linear systems where the coefficient matrix is a square matrix. Determinant of a matrix, Inverse using the classical adjoint method and the Cramer’s rule.
    Week 4:Axiomatic definition of a vector space, examples, subspaces and linear combination and linear span, finite dimensional vector space, fundamental subspaces associated with a matrix
    Week 5:Linear independence and dependence, linear independence and the rank of a matrix, basis of a vector space, constructing a basis of a finite dimensional vector space
    Week 6:Linear transformations, rank-nullity theorem and its application to maps between finite dimensional vector spaces.
    Week 7: Ordered bases, matrix of a linear transformation and similarity of matrices
    Week 8:Dot/Inner product in a vector space, Cauchy Schwartz inequality, angle between two vectors, Projection of a vector onto another vector
    Week 9:Gram-Schmidt orthogonalization process and the QR- decomposition, least square solution of a non-consistent linear system and the orthogonal projections,
    Week 10:Motivation, definition and examples for eigenvalues and eigenvectors, Schur’s unitary triangularization,
    Week 11: Diagonalizable matrices, Criteria for diagonalizability, diagonalizability of Normal matrices, Spectral theorem for Hermitian matrices
    Week 12:Quadratic forms, Sylvester’s law of inertia, Classification of quadrics

    BOOKS AND REFERENCES

    Linear Algebra and its Applications – Gilbert Strang, Cenage Learning
    Linear Algebra – K Hoffman, R Kunze, Pearson
    Linear Algebra: A Geometric Approach – S Kumaresan, PHI

    INSTRUCTOR BIO

    Prof. Arbind Kumar Lal

    IIT Kanpur
    My research interests are in algebraic graph theory.

    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:17 October 2020, 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 Kanpur. It will be e-verifiable at nptel.ac.in/noc
    • Only the e-certificate will be made available. Hard copies will not be dispatched.

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