Linear Algebra, Solutions Manual Ideas and Applications

Linear Algebra: Ideas and Applications, Fourth Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, this book successfully helps readers to understand not only how to implement a technique, but why its use is important.

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 ℝn / 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

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