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! Chapter/Section # !! Title !! Page # | ! Chapter/Section # !! Title !! Page # | ||
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! colspan="3" | Chapter 1: | ! colspan="3" | Chapter 1: DETERMINANTS | ||
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| 1.1 || Number Fields || 1 | | 1.1 || Number Fields || 1 | ||
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| || Problems || 28 | | || Problems || 28 | ||
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! colspan="3" | Chapter 2: | ! colspan="3" | Chapter 2: LINEAR SPACES | ||
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| 1 || | | 2.1 || Definitions || 31 | ||
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| 2 || | | 2.2 || Linear Dependence || 36 | ||
|- | |- | ||
| 2.3 || Bases, Components, Dimension || 38 | |||
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| | | 2.4 || Subspaces || 42 | ||
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| 2 || | | 2.5 || Linear Manifolds || 49 | ||
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| | | 2.6 || Hyperplanes || 51 | ||
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| | | 2.7 || Morphisms of Linear Spaces || 53 | ||
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| || Problems || 56 | |||
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! colspan="3" | | ! colspan="3" | Chapter 3: SYSTEMS OF LINEAR EQUATIONS | ||
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| 1 || | | 3.1 || More on the Rank of a Matrix || 58 | ||
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| 2 || | | 3.2 || Nontrivial Compatibility of a Homogeneous Linear System || 60 | ||
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| 3 || | | 3.3 || The Compatibility Condition for a General Linear System || 61 | ||
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| 4 || | | 3.4 || The General Solution of a Linear System || 63 | ||
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| 3.4 || Geometric Properties of the Solution Space || 65 | |||
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| 3.4 || Methods for Calculating the Rank of a Matrix || 67 | |||
|- | |- | ||
| | | || Problems || 71 | ||
|- | |- | ||
| | ! colspan="3" | Chapter 4: LINEAR FUNCTIONS OF A VECTOR ARGUMENT | ||
|- | |- | ||
| | | 4.1 || Linear Forms || 75 | ||
|- | |- | ||
! colspan="3" | Chapter 6: | | 4.2 || Linear Operators || 77 | ||
|- | |||
| 4.3 || Sums and Products of Linear Operators || 82 | |||
|- | |||
| 4.4 || Corresponding Operations on Matrices || 84 | |||
|- | |||
| 4.5 || Further Properties of Matrix Multiplication || 88 | |||
|- | |||
| 4.6 || The Range and Null Space of a Linear Operator || 93 | |||
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| 4.7 || Linear Operators Mapping a Space \(K_n\) into Itself || 98 | |||
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| 4.8 || Invariant Subspaces || 106 | |||
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| 4.9 || Eigenvectors and Eigenvalues || 108 | |||
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| || Problems|| 113 | |||
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! colspan="3" | Chapter 5: COORDINATE TRANSFORMATIONS | |||
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| 5.1 || Transformation to a New Basis || 118 | |||
|- | |||
| 5.2 || Consecutive Transformations || 120 | |||
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| 5.3 || Transformation of the Components of a Vector || 121 | |||
|- | |||
| 5.4 || Transformation of the Coefficients of a Linear Form || 123 | |||
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| 5.5 || Transformation of the Matrix of a Linear Operator || 124 | |||
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| *5.6 || Tensors || 126 | |||
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| || Problems || 131 | |||
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! colspan="3" | Chapter 6: THE CANONICAL FORM OF THE MATRIX OF A LINEAR OPERATOR | |||
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| 1 || Some standard mappings of the plane || 133 | | 1 || Some standard mappings of the plane || 133 |