Linear Algebra
Teacher
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.
Course 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.
| COs | Course outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 | · Use the basic concepts of vector and matrix algebra, including linear dependence / independence, basis and dimension of a subspace, rank and nullity, for analysis of matrices and systems of linear equations.
|
PO1(3), PO2(2) |
| CO2 | · Evaluate determinants and use them to discriminate between invertible and non-invertible matrices.
|
PO2(2), PO4(2) |
| CO3 | · Solve systems of linear equations using multiple methods, including Gaussian elimination and matrix inversion. Carry out matrix operations, including inverses and determinants.
|
PO9(3) |
| CO5 | · Demonstrate understanding of linear independence, span, and basis.
|
PO2(2), PO1(3) |
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 1:Vector spaces, definition and examples
https://www.youtube.com/watch?v=fNk_zzaMoSs
https://study.com/academy/lesson/vector-spaces-definition-example.html
Session 2: More Problems on Vector space
Session 3: Subspace,definition and examples
Session 4 : Theorems on Subspace
Session 5 : Span of a set.
Session 6: Theorems on Span of a set
https://www.slideshare.net/ssatul99/independence-basis-and-dimension
Session 7 : More about Subspace
Session 8 : (2 hour)
Project-1: Preaperation of detail report on Vector Space and Subspace.
Session 9 : (2 hr)
Project-2: To show that Span of a subset of a vector space is a Vector Space.
Session 10 : Direct Sum
Session 11 : Linear dependence and independence: definitions with examples
Session 12 : Dimension and basis,Theorems on Dimensions and Basis
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
Session 17 : Range and kernel
Session 18 : Rank and nullity
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
Session 23: Theorems on Composition of Linear maps
Session 24:Matrix associated with linear map
Session 25: Linear Map Associated with Matrix
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
Session 29 : Theorems on Rank and Nulity
Session 30: Theorems on Transpose of a matrix.
Session 31: Elementary row operations
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
Session 35: (2 hr)
Practice-2 : Matrix Multiplication,Transpose and Inversion.
Session 36 : Systems of linear equations
Session 37: Determinants, Minors
Session 38 : Rank of a matrix by determinant Method
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.
Session 41 : (2 hr )
Practice-4: Solution of AX=B using Gauss elimination method.
Session 42: Product of determinants and Theorems on Product of determinants
Session 43: Application to linear equations and existence of solution
Session 44 : Eigen value and Eigen vector.
Session 45 :More Problems on Eigen value and eigen Vector
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
Session 48 : (2 hr)
Practice-6: Finding Eigen values and Eigen Vector of a matrix
Case Studies
Case Studies
Our Main Teachers
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 […]