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Chapter Ⅰ. Linear Algebra 1

1. Definitions and Examples of Vector Spaces 1

2. Bases and Dimension 8

3. Linear Transformations 17

4. Products and Direct Sums 30

5. Quotient Spaces and the Isomorphism Theorems 38

6. Duals and Adjoints 46

7. Symmetric Bilinear Forms 53

Chapter Ⅱ. Multilinear Algebra 59

1. Multilinear Maps and Tensor Products 59

2. Functorial Properties of Tensor Products 68

3. Alternating Maps and Exterior Powers 83

4. Symmetric Maps and Symmetric Powers 94

Chapter Ⅲ. Canonical Forms of Matrices 98

1. Preliminaries on Fields 98

2. Minimal and Characteristic Polynomials 105

3. Eigenvalues and Eigenvectors 117

4. The Jordan Canonical Form 132

5. The Real Jordan Canonical Form 141

6. The Rational Canonical Form 159

Chapter Ⅳ. Normed Linear Vector Spaces 171

1. Basic Definitions and Examples 171

2. Product Norms and Equivalence 180

3. Sequential Compactness and the Equivalence of Norms 186

4. Banach Spaces 200

Chapter Ⅴ. Inner Product Spaces 206

1. Real Inner Product Spaces 206

2. Self-adjoint Transformations 221

3. Complex Inner Product Spaces 236

4. Normal Operators 243

Glossary of Notation 254

References 259

Subject Index 261