Linear Algebra

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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

Session 4 : Theorems on Subspace

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 11 : Linear dependence and independence: definitions with examples

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

Session 16 : Theorems on Linear Transformation

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