Concise Introduction to Linear Algebra

Concise Introduction to Linear Algebra deals with the subject of linear algebra, covering vectors and linear systems, vector spaces, orthogonality, determinants, eigenvalues and eigenvectors, singular value decomposition. It adopts an efficient approach to lead students from vectors, matrices quickly into more advanced topics including, LU decomposition, orthogonal decomposition, Least squares solutions, Gram-Schmidt process, eigenvalues and eigenvectors, diagonalizability, spectral decomposition, positive definite matrix, quadratic forms, singular value decompositions and principal component analysis. This book is designed for onesemester teaching to undergraduate students. Cover -- Half title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Chapter 1 Vectors and linear systems -- 1.1 Vectors and linear combinations -- 1.2 Length, angle and dot products -- 1.3 Matrices -- Chapter 2 Solving linear systems -- 2.1 Vectors and linear equations -- 2.2 Matrix operations -- 2.3 Inverse matrices -- 2.4 LU decomposition -- 2.5 Transpose and permutation -- Chapter 3 Vector spaces -- 3.1 Spaces of vectors -- 3.2 Nullspace, row space and column space -- 3.3 Solutions of Ax = b -- 3.4 Rank of matrices -- 3.5 Bases and dimensions of general vector spaces -- Chapter 4 Orthogonality -- 4.1 Orthogonality of the four subspaces -- 4.2 Projections -- 4.3 Least squares approximations -- 4.4 Orthonormal bases and Gram-Schmidt -- Chapter 5 Determinants -- 5.1 Introduction to determinants -- 5.2 Properties of determinants -- Chapter 6 Eigenvalues and eigenvectors -- 6.1 Introduction to eigenvectors and eigenvalues -- 6.2 Diagonalizability -- 6.3 Applications to differential equations -- 6.4 Symmetric matrices and quadratic forms -- 6.5 Positive definite matrices -- Chapter 7 Singular value decomposition -- 7.1 Singular value decomposition -- 7.2 Principal component analysis -- Chapter 8 Linear transformations -- 8.1 Linear transformation and matrix representation -- 8.2 Range and null spaces of linear transformation -- 8.3 Invariant subspaces -- 8.4 Decomposition of vector spaces -- 8.5 Jordan normal form -- 8.6 Computation of Jordan normal form -- Chapter 9 Linear programming -- 9.1 Extreme points -- 9.2 Simplex method -- 9.3 Simplex tableau -- Index