Linear Algebra and Vector Calculus (2110015) Gujarat Technological University 2017

740 Pages·2017·84.985 MB·English

by  Ravish R Singh;Mukul Bhatt

Linear Algebra and
Vector Calculus
(2110015)
Gujarat Technological University 2017
Fourth Edition
About the Authors
Ravish R Singh is presently Academic Advisor at Thakur
Educational Trust, Mumbai. He obtained a BE degree from
University of Mumbai in 1991, an MTech degree from IIT
Bombay in 2001, and a PhD degree from Faculty of Technology,
University of Mumbai, in 2013. He has published several books
with McGraw Hill Education (India) Private Limited on varied
subjects like Engineering Mathematics (I and II), Applied
Mathematics, Electrical Engineering, Electrical and Electronics
Engineering, etc., for all-India curricula as well as regional
curricula of some universities like Gujarat Technological University, Mumbai University,
Pune University, Jawaharlal Nehru Technological University, Anna University,
Uttarakhand Technical University, and Dr A P J Abdul Kalam Technical University
(formerly known as UPTU). Dr Singh is a member of IEEE, ISTE, and IETE, and has
published research papers in national and international journals. His fields of interest
include Circuits, Signals and Systems, and Engineering Mathematics.
Mukul Bhatt is presently Assistant Professor, Department of
Humanities and Sciences, at Thakur College of Engineering
and Technology, Mumbai. She obtained her MSc (Mathematics)
from H N B Garhwal University in 1992. She has published
several books with McGraw Hill Education (India) Private
Limited on Engineering Mathematics (I and II) and Applied
Mathematics for all-India curricula as well as regional curricula
of some universities like Gujarat Technological University,
Mumbai University, Pune University, Jawaharlal Nehru
Technological University, Anna University, Uttarakhand
Technical University, and Dr A P J Abdul Kalam Technical University (formerly
known as UPTU). She has seventeen years of teaching experience at various levels in
engineering colleges in Mumbai and her fields of interest include Integral Calculus,
Complex Analysis, and Operation Research. She is a member of ISTE.
Linear Algebra and
Vector Calculus
(2110015)
Gujarat Technological University 2017
Fourth Edition
Ravish R Singh
Academic Advisor
Thakur Educational Trust
Mumbai, Maharashtra
Mukul Bhatt
Assistant Professor
Department of Humanities and Sciences
Thakur College of Engineering and Technology
Mumbai, Maharashtra
McGraw Hill Education (India) Private Limited
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To
Our parents-in-law
Shri Uday Pratap Singh
Shrimati Shaila Singh
Ravish R Singh
Late Shri Dataram Bhatt
Shrimati Sateshwaridevi Bhatt
Mukul Bhatt

contents
O
Preface xi
1. MAtrices And systeMs of LineAr
equAtions 1.1-1.102
1.1 Introduction 1.1
1.2 Matrix 1.1
1.3 Some Definitions Associated with Matrices 1.1
1.4 Some Special Matrices 1.4
1.5 Elementary Transformations 1.19
1.6 System of Non-Homogeneous Linear Equations 1.25
1.7 System of Homogeneous Linear Equations 1.47
1.8 Inverse of a Matrix 1.61
1.9 Rank of a Matrix 1.79
1.10 Applications of Systems of Linear Equations 1.97
2. Vector spAces 2.1–2.118
2.1 Introduction 2.1
2.2 Euclidean Vector Space 2.1
2.3 Vector Spaces 2.11
2.4 Subspaces 2.17
2.5 Linear Combination 2.23
2.6 Span 2.30
2.7 Linear Dependence and Independence 2.47
2.8 Basis 2.63
2.9 Finite Dimensional Vector Space 2.69
2.10 Basis and Dimension for Solution Space of the Homogeneous
Systems 2.70
2.11 Reduction and Extension to Basis 2.78
2.12 Coordinate Vector Relative to a Basis 2.87
2.13 Change of Basis 2.90
2.14 Row Space, Column Space and Null Space 2.103
2.15 Rank and Nullity 2.115
3. LineAr trAnsforMAtion 3.1–3.90
3.1 Introduction 3.1
3.2 Euclidean Linear Transformation 3.1
3.3 Linear Transformations 3.2
3.4 Linear Operators (Types of Linear Transformations) 3.3
3.5 Linear Transformations from Images of Basis Vectors 3.12
3.6 Composition of Linear Transformation 3.16
3.7 Kernel (Null Space) and Range of a Linear Transformation 3.24
3.8 Inverse Linear Transformations 3.46
3.9 The Matrix of a Linear Transformation 3.55
3.10 Effect of Change of Bases on Linear Operators 3.77
3.11 Similarity of Matrices 3.86
4. inner product spAces 4.1–4.52
4.1 Introduction 4.1
4.2 Inner Product Spaces 4.1
4.3 Orthogonal and Orthonormal Basis 4.24
4.4 Gram–Schmidt Process 4.25
4.5 Orthogonal Complements 4.39
4.6 Orthogonal Projection 4.43
4.7 Least Squares Approximation 4.47
5. eigenVALues And eigenVectors 5.1–5.103
5.1 Introduction 5.1
5.2 Eigenvalues and Eigenvectors 5.1
5.3 Cayley–Hamilton Theorem 5.28
5.4 Similarity of Matrices 5.38
5.5 Diagonalization 5.38
5.6 Quadratic Form 5.64
5.7 Conic Sections 5.95
6. Vector functions 6.1–6.81
6.1 Introduction 6.1
6.2 Vector Function of a Single Scalar Variable 6.1
6.3 Tangent, Normal and Binormal Vectors 6.2
6.4 Arc Length 6.15
6.5 Curvature and Torsion 6.17
6.6 Scalar and Vector Point Function 6.24
6.7 Gradient 6.25
6.8 Divergence 6.46
6.9 Curl 6.48
6.10 Properties of Gradient, Divergence and Curl 6.60
6.11 Second Order Differential Operator 6.65
7. Vector cALcuLus 7.1–7.87
7.1 Introduction 7.1
7.2 Line Integrals 7.1
7.3 Path Independence of Line Integrals
(Conservative Field and Scalar Potential) 7.2
7.4 Green’s Theorem in the Plane 7.16
7.5 Surface Integrals 7.33
7.6 Volume Integrals 7.41
7.7 Gauss’ Divergence Theorem 7.44
7.8 Stokes’ Theorem 7.62
Appendix 1 Integral Formulae A1.1– A1.2
Solved Question Paper (May/June-2012) SQP.1–SQP.25
Solved Question Paper (Summer-2013) SQP.2– SQP.17
Solved Question Paper (Summer-2014) SQP.3– SQP.33
Solved Question Paper (Winter-2014) SQP.1– SQP.27
Solved Question Paper (Summer-2015) SQP.1– SQP.26
Index I.1–I.4