Elementary Linear Algebra

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