Linear Algebra Step by Step

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1 Linear Equations and Matrices 1

1.1 Systems of Linear Equations 1

1.2 Gaussian Elimination 12

1.3 Vector Arithmetic 27

1.4 Arithmetic of Matrices 41

1.5 Matrix Algebra 57

1.6 The Transpose and Inverse of a Matrix 75

1.7 Types of Solutions 91

1.8 The Inverse Matrix Method 105

Des Higham Interview 127

2 Euclidean Space 129

2.1 Properties of Vectors 129

2.2 Further Properties of Vectors 143

2.3 Linear Independence 159

2.4 Basis and Spanning Set 171

Chao Yang Interview 190

3 General Vector Spaces 191

3.1 Introduction to General Vector Spaces 191

3.2 Subspace of a Vector Space 202

3.3 Linear Independence and Basis 216

3.4 Dimension 229

3.5 Properties of a Matrix 239

3.6 Linear Systems Revisited 254

Janet Drew Interview 275

4 Inner Product Spaces 277

4.1 Introduction to Inner Product Spaces 277

4.2 Inequalities and Orthogonality 290

4.3 Orthonormal Bases 306

4.4 Orthogonal Matrices 321

Anshul Gupta Interview 338

5 Linear Transformations 339

5.1 Introduction to Linear Transformations 339

5.2 Kernel and Range of a Linear Transformation 352

5.3 Rank and Nullity 364

5.4 Inverse Linear Transformations 372

5.5 The Matrix of a Linear Transformation 389

5.6 Composition and Inverse Linear Transformations 407

Petros Drineas Interview 429

6 Determinants and the Inverse Matrix 431

6.1 Determinant of a Matrix 431

6.2 Determinant of Other Matrices 439

6.3 Properties of Determinants 455

6.4 LU Factorization 472

Françoise Tisseur Interview 490

7 Eigenvalues and Eigenvectors 491

7.1 Introduction to Eigenvalues and Eigenvectors 491

7.2 Properties of Eigenvalues and Eigenvectors 503

7.3 Diagonalization 518

7.4 Diagonalization of Symmetric Matrices 533

7.5 Singular Value Decomposition 547

Brief Solutions 567

Index 605