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These notes are from my iPad mostly using GoodNotes; for some of the special topics at the bottom, I used Noteshelf.
Day 1 -Motivation and Set Theory
Day 2 -Notation and Techniques of Proof
Day 3 -Groups and Fields; More Techniques of Proof
Day 4 -Vector Spaces: Definition and Examples
Day 5 -Subspaces: Definition, Examples, and the Subspace Test
Day 6 -Linear Combinations and Span
Day 7 -Spanning Sets, Linear Independence, and Bases
Day 8 -Examples of Bases; The Hausdorff Maximality Principle
Day 9 -Proof That Every Vector Space Has a Basis
Day 10 -Dimension: Definition and Proof That it is Well-Defined
Day 10.1 Linear Equations and Matrices, Introduction
Day 10.2 Pivots and bases; Proof that Dimension is Well-Defined in Finite Dimensional Spaces
Day 10.3 Sizes of Spanning and Linearly Independent Sets in Finite Dimensional Spaces; Invertibility
Day 11 -More on Dimension; Norms: Definition and Examples
Day 12 -More Examples of Norms; Definition of Inner Product
Day 13 -Inner Products: Definition and Examples
Day 14 -Norms From Inner Products and the Parallelogram Property
Day 15 -Orthogonality and Orthonormality; the Gram-Schmidt Process
Day 16 -Proof of Gram-Schmidt; Orthogonal Complements
Day 17 -Orthogonal Decomposition and Projections
Day 18 -Linear Transformations: Definition and Examples
Day 19 -More Examples of Linear Transformations; Isomorphisms; Reduction of Finite Dimensional Spaces to F^n
Day 20 -Reduction of Linear Transformations Between Finite Dimensional Spaces to Matrices; Change of Basis
Day 21 -Operations for Linear Transformations and Notation
Day 22 -Invertibility; Reduction to Finite-Dimensional Spaces; Matrix Form for Linear Transformations and Matrix Product
Day 23 -Matrix Norms: Definition and Examples; Diagonalizability; Eigenvalues and Eigenspaces
Day 24 -Similarity and Permutations
Day 25 -Definition of the Determinant and Properties
Day 26 -More Determinant Properties
Day 27 -Characteristic Polynomial; Linear Indepence of Eigenvectors; Failure of Diagonalizability
Day 28 -Properties Equivalent to Diagonalizability; Semi-Simplicity; the Adjoint of a Matrix and Normality
Day 29 -(Unitary) Diagonalizability of Normal Matrices; Unitary and Self-Adjoint Matrices
Day 30 -Eigenvalues of Unitary and Self-Adjoint Matrices; Beginning of Proof Unitary Triangularization of Arbitrary Matrices
Day 31 - Unitary Triangularization of Arbitrary Matrices; Absolute Value and Polar Decomposition of Matrices
Day 32 -Proof of Polar Decomposition
Day 32.1 Nilpotent Matrices: Definition and Properties
Day 32.2 Jordan Canonical Form For Nilpotent Matrices
Day 32.4 Linear Functionals; Definition of Dual Space; Finite-Dimensional Isomorphisms of a Vector Space and its Dual
Day 32.5 Continuous Duals and Double Duals
Day 32.6 Riesz Representation Theorem; Definition and Existence of Adjoints; The Matrix of an Adjoint (all in finite dimensions)
Day 32.7 Direct (Cross) Product of Vector Spaces; Multilinear Maps
Day 32.8 Definition of Multilinear Maps and Tensor Products
Day 32.9 Dimension of a Tensor Product; Definition of Simple Tensor
Day 32.91 Tensor Products of Linear Maps; Matrix Representations of Tensor Products
