Linear Algebra

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Dr Tumbanath Samantara

Category

Core Courses

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Course Name : Linear Algebra

Code(Credit) : CUTM1512(3-1-2)

Course Objectives

This course unit aims to introduce the basic ideas and techniques of linear algebra for use in many other lecture courses.

Solve systems of linear equations using various methods including Gaussian and Gauss Jordan elimination and inverse matrices.

Proof of basic results in linear algebra using appropriate proof-writing techniques.

Learning Outcomes

Upon successful completion of this course, students will be able to:

Use the basic concepts of vector and matrix algebra, including linear dependence / independence, basis and dimension of a vector space, rank and nullity, for analysis of matrices and systems of linear equations.

Evaluate determinants and use them to discriminate between invertible and non-invertible matrices.

Solve systems of linear equations using multiple methods, including Gaussian elimination and matrix inversion. Carry out matrix operations, including inverses and determinants.

Demonstrate understanding of linear independence, span, and basis.

Course Syllabus

Module I

Vector spaces, definition and examples,Subspace,Span of a set.

Project-1: Preparation of detail report on Vector space,Subspace

Project-2: To show that span of a subset of a vector space is a Vector space

Module II
Linear dependence and independence,Dimension and basis

Project-3: Preparation of detail report on dependency of vectors .

Project-4: Finding dimension and basis of a vector Space.

Module III
Linear transformation, definition and examples, Range and kernel, Rank and nullity, The space L(U,V )

Project-5:To find out Range and Kernel of different  Linear Transformations.

Project-6: Establishment of R(T) and N(T) are sub spaces of V and U respectively
in the linear map T:U-->V

Module IV
Composition of Linear maps,Matrix and linear map,linear operations,matrix multiplication

Project-7: To show that L(U,V) is a vector space

Project-8: Establish the condition of non Singularity of Linear Map.

Module V
Rank and nullity of matrix,Transpose of a matrix,Elementary row operations

Project-9: Preparation of working rule for finding rank of a matrix with
example.

Project-10:To find the matrix associated  with Linear Maps

Practice 1 : Introduction to MATLAB. Matrix addition and multiplication

Practice 2 : Matrix Transpose , Matrix Inversion

Module VI
Systems of linear equations,Matrix inversion,Determinants, minors ,Rank of a matrix

Project-11: Finding rank of a matrix by Determinant Method

Practice 3: Finding Rank of a Matrix

Practice 4: Solution of AX=B using Gauss Elimination method .

Module VII
Product of determinants,Application to linear equations,Eigen value and Eigen vector

Project-12: Finding the eigen value and eigen vector of a 3x3 matrix having repeated root in it's characteristic equation.

Practice 5 : Solution of AX=B using, Gauss Seidal and Gauss-Jacobi method

Practice 6 : Finding Eigen Values and Eigen vectors of Matrix

Text Book:

An Introduction to Linear Algebra by V. Krishnamurty,V.P.Mainra, J.L.Arora, Affiliated East-West pressPvt.Ltd.

Chapters: 3,4 (4.1 to 4.7), 5,6 (6.5 to 6.8)

Reference Books:

Basic Structures in Algebra, Part-I : J.N. Patnaik

Matrix Theory and Linear Algebra : I.N. Herstein and D.J. Winter (Ma Chilan Publishing company)

First course in Linear algebra : Bhattacharya, Jain and Nagpaul (New Age International)

Session Plan

Session 2: More Problems on Vector space

https://www.youtube.com/watch?v=tfSOQL1m7f0

Session 4 : Theorems on Subspace

Session 7 :  More about Subspace

https://www.youtube.com/watch?v=TuOQ4-kmcE0

Session 8 : (2 hour)

Project-1: Preaperation of detail report on Vector Space and Subspace.

https://www.youtube.com/watch?v=TuOQ4-kmcE0

Session 9 : (2 hr)

Project-2: To show that Span of a subset of a vector space is a Vector Space.

Session 11 : Linear dependence and independence: definitions with examples

https://www.youtube.com/watch?v=YrHlHbtiSM0

https://www.youtube.com/watch?v=OFALIHBY5Bw

Session 12 : Dimension and basis,Theorems on Dimensions and Basis

https://slideplayer.com/slide/5347557/

Session 13: (2hr)

Project-3: Preparation of detail report on dependency of vectors.

Session 14: (2hr)

Project-4: Finding dimension and basis of a vector space

Session 15 :  Linear transformation, definition and examples

https://www.youtube.com/watch?v=kYB8IZa5AuE&list=PLMIiGh1RmKQXgfeU8AdYiR329HFQbTjpA&index=1

Session 16 : Theorems on Linear Transformation

https://www.youtube.com/watch?v=CISI-NOU6zE

Session 19 : The space L(U,V )

Session 20 : (2hr)

Project-5: To find out Range and Kernel of different Linear Transformation

Session 21 : (2hr)

Project-6: Establishment of R(T) and N(T) are subspace of V and U respectively
in the linear map T:U->V

Session 22: Composition of Linear maps

https://www.youtube.com/watch?v=AfjRc-IxZQk

Session 23: Theorems on Composition of Linear maps

https://www.youtube.com/watch?v=gLtl7qg7-XU

Session 24:Matrix associated with linear map

https://www.youtube.com/watch?v=LlbzPlS3ZQk

Session 25: Linear Map Associated with Matrix

https://www.youtube.com/watch?v=6pQGDsEUJ1I

Session 26 : (2 hr)

Project-7: To show that  L(U,V) is a vector space

Session 27 : (2 hr)

Project-8: Establish the condition of non Singularity of Liner map

Session 28: Rank and nullity of matrix

https://www.youtube.com/watch?v=HBiYU8gR_X4

Session 29 : Theorems on Rank and Nulity

https://www.youtube.com/watch?v=HBiYU8gR_X4

Session 30: Theorems on Transpose of a matrix.

https://www.youtube.com/watch?v=OG2Wbkd4vgU

Session 31: Elementary row operations

https://www.youtube.com/watch?v=ICtDaMWTrrg

Session 32: (2 hr)

Project-9 : Preparation of working rule for finding rank of a matrix with example.

Session 33: (2 hr)

Project-10 : To find the matrix associated with Linear Maps

Session 34: (2 hr )

Practice-1: Introduction to MATLAB,Addition of Matrix

https://www.youtube.com/watch?v=CnuFGYBvnmU

Session 35:  (2 hr)

Practice-2 : Matrix Multiplication,Transpose and Inversion.

https://www.youtube.com/watch?v=CnuFGYBvnmU

Session 36 : Systems of linear equations

https://www.youtube.com/watch?v=B7XtnvNwEPA

Session 37: Determinants, Minors

https://www.youtube.com/watch?v=55AoWKZZtww

Session 38 : Rank of a matrix by determinant Method

https://www.youtube.com/watch?v=5oDdSb9Jv6c

Session 39 : (2 hr )

Project-11: Finding rank of a matrix by determinant method.

Session 40 : (2 hr )

Practice -3: Finding rank of a matrix.

https://www.youtube.com/watch?v=Y3G2VpWCvd4

Session 41 : (2 hr )

Practice-4: Solution of AX=B using Gauss elimination method.

https://www.youtube.com/watch?v=JY99xfMgwnk

Session 42: Product of determinants and Theorems on Product of determinants

https://www.youtube.com/watch?v=9CqBrhLkzE8

Session 43: Application to linear equations and existence of solution

https://www.youtube.com/watch?v=gjQD5uKe9kE

Session 44 : Eigen value and Eigen vector.

https://www.youtube.com/watch?v=ue3yoeZvt8E

Session 45 :More Problems on Eigen value and eigen Vector

https://www.youtube.com/watch?v=cHOsd2PhkqE&vl=en

Session 46 : (2 hr )

Project-12: Finding the Eigen value ad Eigen Vector of a 3x3 matrix having repeated roots in characteristic equation.

Session 47 : (2 hr )

Practice-5 : Solution of AX=B using Gauss-Seidal and Gauss-Jacobi Method

https://www.youtube.com/watch?v=YwGre83-zAM

Session 48 : (2 hr)

Practice-6: Finding Eigen values and Eigen Vector of a matrix

https://www.youtube.com/watch?v=jhsLnYfsNdU

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    Dr. Tumbanath  Samantara currently works as Associate Professor & HoD, Department of Applied Mathematics, , School of Applied Sciences, Centurion University of Technology and Management, (Bhubaneswar Campus) Odisha. He has completed his Ph.D. in the field of Fluid Dynamics. He has more than 10 years of research experience and 19 years of teaching experience. During […]