MATH 5101: Linear Mathematics in Finite Dimensions

Chapter 1 VECTOR SPACES 1.1 Lecture 1 (Wednesday) 1.1.1 Three archetypical equations of linear algebra 1.1.2 Vector as an aggregate of entities 1.1.3 Vector space: Definition and Examples 1.1.4 Subspace of a vector space 1.2 Lecture 2 (Friday) 1.2.1 The Subspace Theorem 1.2.2 Spanning set 1.2.3 Linear independence (material for next lecture) 1.2.4 Basis; coordinates (material for next lecture) 1.2.5 Basis-induced isomorphism (material for next lecture) 1.3 Lecture 3 (Monday) 1.3.1 Spanning set example 1.3.2 Linear independence 1.3.3 Basis; coordinates 1.3.4 Basis-induced isomorphism 6 1.4 Lecture 4 (Wednesday) 1.4.1 Coordinate representative of a vector relative to a given basis (Existence and uniqueness) 1.4.2 Coordinates 1.5 Lecture 5 (Friday) 1.5.1 Basis-induced correspondence between V and Rp as structure preserving 1.5.2 Preservation of linear independence and dependence of a set of sets of vectors 1.5.3 Isomorphism and its basis independence 1.6 Lecture 6 (Wednesday) 1.6.1 Structure preservation 1.6.2 Dimension of V : its coordinate invariance 1.6.3 Mutual implication of linear independence and spanning property (in an appropriate context) 1.7 Lecture 7 (Friday) 1.7.1 Linear functions on a vector space V 1.7.2 The dual vector space V ∗ 1.7.3 Dirac’s bracket notation 1.7.4 Basis (and their coordinate lines) vs. dual basis (and their coordinate surfaces); vectors vs. covectors 1.8 Lecture 8 (Monday) 1.8.1 The Duality Principle 1.8.2 Example for Rn : {Column vectors}=V ; {Row vectors}=V ∗ 1.8.3 Addition of vectors and covectors 7 1.8.4 APPENDIX for Lecture 8: Four Important Examples of Linear Functions: (i) Dollar Values of Fruit Inventories (ii) Interpolation of sampled data: Lagrangian interpolation via quadratic polynomials (iii) Interpolation of sampled data via Roof functions (iv) Interpolation of sampled data via Band-limited basis functions 1.9 Lecture 9 (Wednesday) 1.9.1 Bilinear functional 1.9.2 Metric as an inner product 1.9.3 Metric as a natural isomorphism between V and V ∗ 1.10 Lecture 10 (Friday) 1.10.1 Reciprocal vector basis for V 1.10.2 Its relation to the dual basis for V ∗ 1.10.3 Geometrical relation between vectors and their linear functionals in V ∗ 8 Chapter 2 LINEAR TRANSFORMATIONS 2.1 Lecture 11 (Monday) 2.1.1 Transformations: who needs them? 2.1.2 Linear transformations: their geometric and algebraic definition 2.1.3 Onto and one-to-one transformations 2.1.4 Null Space and Range Space: their importance 2.1.5 Example 2.2 Lecture 12 (Wednesday) 2.2.1 T-induced basis properties 2.2.2 Dimensional consistency 2.2.3 Direct sum of two vector spaces 2.3 Lecture 13 (Friday) 2.3.1 Operations with linear transformations Sum of two transformations Composition of linear transformations Inverse of an invertible transformation