LINEAR ALGEBRA – MATHEMATICS : PAPER I (1) – MTC 121 – Golden Series

1 Vector Spaces

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Vectors spaces and subspaces: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Null spaces, Column spaces and Linear transformation . . . . . . . . . . . . . . . . . 13
1.4 Linearly independent sets and Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 The dimension of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 Eigenvalues and Eigenvectors

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.2 Eigenvalues and Eigenvectors: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3 The Characteristic Equation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.5 Eigenvectors and linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3 Orthogonality and Symmetric Matrices

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 Inner Product, Length and Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Orthogonal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Orthogonal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Diagonalization of Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.6 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 The Geometry of Vector Spaces

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.2 Affine Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Affine independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4 Convex Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

About the Book :

This book is based on a course Linear Algebra. We have written this book as per the revised syllabus of F.Y. B.Sc.(Computer Science) Mathematics, revised by Savitribai Phule Pune University, Pune, implemented from June 2019. Linear Algebra is the most useful subject in all branches of mathematics and it is used extensively in applied mathematics and Computer Graphics.

Linear Algebra is the study of vector spaces, which are mathematical structures used to design aircraft models. It is a bridge connecting mathematics with various branches of computer science. We study, how problems in almost every conceivable discipline can be solved using techniques of Linear Algebra. The aim of this textbook on Linear Algebra is to introduce basic concepts and some applications to model computational problems.

In Chapter 1, we explain the concepts of vector spaces, subspaces , Null spaces, Row spaces and
Column spaces. They are illustrated with the help of a number of useful and interesting examples.The
dimension of a vector space and rank of a matrix is explained at the end of the chapter.
Chapter 2 deals with Eigenvalues, Eigenvectors,characteristic equations and diagonalization process.
The connection of eigenvectors and linear transformations is also explained with the help of examples.
Orthogonality of vectors is the core part of chapter 3. The concepts of inner product,length and
orthogonal projections is discussed with reference to n-dimensional real vector spaces.The quadratic
forms of symmetric matrices is introduces at the end of this chapter.
The fourth chapter focusses on the geometric visualization of real vector spaces with the help of
affine combination of vectors and convex sets.