STUDENT MANUAL 1 1 SYSTEMS OF LINEAR EQUATIONS 3 1.1 The Vector Space of m × n Matrices / 3 1.1.2 Applications to Graph Theory I / 7 1.2 Systems / 8 1.2.2 Applications to Circuit Theory / 11 1.3 Gaussian Elimination / 13 1.

3.2 Applications to Traffic Flow / 18 1.4 Column Space and Nullspace / 19 2 LINEAR INDEPENDENCE AND DIMENSION 26 2.1 The Test for Linear Independence / 26 2.2 Dimension / 33 2.2.2 Applications to Differential Equations / 37 2.3 Row Space and the Rank-Nullity Theorem / 38 3 LINEAR TRANSFORMATIONS 43 3.

1 The Linearity Properties / 43 3.2 Matrix Multiplication (Composition) / 49 3.2.2 Applications to Graph Theory II / 55 3.3 Inverses / 55 3.3.2 Applications to Economics / 60 3.4 The LU Factorization / 61 3.

5 The Matrix of a Linear Transformation / 62 4 DETERMINANTS 67 4.1 Definition of the Determinant / 67 4.2 Reduction and Determinants / 69 4.2.1 Volume / 72 4.3 A Formula for Inverses / 74 5 EIGENVECTORS AND EIGENVALUES 76 5.1 Eigenvectors / 76 5.1.

2 Application to Markov Processes / 79 5.2 Diagonalization / 80 5.2.1 Application to Systems of Differential Equations / 82 5.3 Complex Eigenvectors / 83 6 ORTHOGONALITY 85 6.1 The Scalar Product in ?n / 85 6.2 Projections: The Gram-Schmidt Process / 87 6.3 Fourier Series: Scalar Product Spaces / 89 6.

4 Orthogonal Matrices / 92 6.5 Least Squares / 93 6.6 Quadratic Forms: Orthogonal Diagonalization / 94 6.7 The Singular Value Decomposition (SVD) / 97 6.8 Hermitian Symmetric and Unitary Matrices / 98 7 GENERALIZED EIGENVECTORS 100 7.1 Generalized Eigenvectors / 100 7.2 Chain Bases / 104 8 NUMERICAL TECHNIQUES 107 8.1 Condition Number / 107 8.

2 Computing Eigenvalues / 108.