Elementary Linear Algebra

This book will probably be unlike any textbook you have ever used and it has been specifically written to facilitate your learning in elementary linear algebra. It has been my experience over more than forty five years of teaching linear algebra that student success is highly correlated with mastery of the definitions and concepts of which there are certainly more then fifty and very likely as many as one hundred. 1 Linear Equations 3 1.1 Linear Equations and Their Solution . . . . . . . . . . . . . . . . . . 3 1.2 Matrices and Echelon Forms . . . . . . . . . . . . . . . . . . . . . . 32 1.3 How to Use it: Applications of Linear Systems . . . . . . . . . . . . 62 2 The Vector Space Rn 91 2.1 Introduction to Vectors: Linear Geometry . . . . . . . . . . . . . . . 91 2.2 Vectors and the Space R n . . . . . . . . . . . . . . . . . . . . . . . . 116 2.3 The Span of a Sequence of Vectors . . . . . . . . . . . . . . . . . . . 140 2.4 Linear independence in Rn . . . . . . . . . . . . . . . . . . . . . . . 165 2.5 Subspaces and Bases of Rn . . . . . . . . . . . . . . . . . . . . . . . 194 2.6 The Dot Product in Rn . . . . . . . . . . . . . . . . . . . . . . . . . 223 3 Matrix Algebra 251 3.1 Introduction to Linear Transformations and Matrix Multiplication . . 251 3.2 The Product of a Matrix and a Vector . . . . . . . . . . . . . . . . . . 273 3.3 Matrix Addition and Multiplication . . . . . . . . . . . . . . . . . . . 294 3.4 Invertible Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 3.5 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 340 3.6 The LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3.7 How to Use It: Applications of Matrix Multiplication . . . . . . . . . 379 4 Determinants 403 4.1 Introduction to Determinants . . . . . . . . . . . . . . . . . . . . . . 403 4.2 Properties of Determinants . . . . . . . . . . . . . . . . . . . . . . . 421 4.3 The Adjoint of a Matrix and Cramer’s Rule . . . . . . . . . . . . . . 447 5 Abstract Vector Spaces 461 5.1 Introduction to Abstract Vector Spaces . . . . . . . . . . . . . . . . . 461 5.2 Span and Independence in Vector Spaces . . . . . . . . . . . . . . . . 478 5.3 Dimension of a finite generated vector space . . . . . . . . . . . . . . 510 5.4 Coordinate vectors and change of basis . . . . . . . . . . . . . . . . . 529 5.5 Rank and Nullity of a Matrix . . . . . . . . . . . . . . . . . . . . . . 551 5.6 Complex Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . 573 5.7 Vector Spaces Over Finite Fields . . . . . . . . . . . . . . . . . . . . 600 5.8 How to Use it: Error Correcting Codes . . . . . . . . . . . . . . . . . 615 6 Linear Transformations 635 6.1 Introduction to Linear Transformations on Abstract Vector Spaces . . 635 6.2 Range and Kernel of a Linear Transformation . . . . . . . . . . . . . 656 6.3 Matrix of a Linear Transformation . . . . . . . . . . . . . . . . . . . 680 7 Eigenvalues and Eigenvectors 703 7.1 Introduction to Eigenvalues and Eigenvectors . . . . . . . . . . . . . 703 7.2 Diagonalization of Matrices . . . . . . . . . . . . . . . . . . . . . . 727 7.3 Complex Eigenvalues of Real Matrices . . . . . . . . . . . . . . . . . 753 7.4 How to Use It: Applications of Eigenvalues and Eigenvectors . . . . . 780 8 Orthogonality in R n 803 8.1 Orthogonal and Orthonormal Sets in Rn . . . . . . . . . . . . . . . . 803 8.2 The Gram-Schmidt Process and QR-Factorization . . . . . . . . . . 824 8.3 Orthogonal Complements and Projections . . . . . . . . . . . . . . . 843 8.4 Diagonalization of Real Symmetric Matrices . . . . . . . . . . . . . . 869 8.5 Quadratic Forms, Conic Sections and Quadratic Surfaces . . . . . . . 895 8.6 How to Use It: Least Squares Approximation . . . . . . . . . . . . . 932