Contents Preface xi Note to the Reader xv Chapter 1. Preliminaries 1 1.1. Vector Spaces 1 1.2. Bases and Coordinates 3 1.3. Linear Transformations 3 1.4. Matrices 4 1.5. The Matrix of a Linear Transformation 5 1.6. Change of Basis and Similarity 6 1.7. Transposes 8 1.8. Special Types of Matrices 8 1.9. Submatrices, Partitioned Matrices, and Block Multiplication 9 1.10. Invariant Subspaces 10 1.11. Determinants 11 1.12. Tensor Products 13 Exercises 14 Chapter 2. Inner Product Spaces and Orthogonality 17 2.1. The Inner Product 17 2.2. Length, Orthogonality, and Projection onto a Line 18 2.3. Inner Products in Cn 21 2.4. Orthogonal Complements and Projection onto a Subspace 23 2.5. Hilbert Spaces and Fourier Series 27 2.6. Unitary Tranformations 31 v
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