📚 Syllabus
Chapter 1: Systems of Linear Equations
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§ 1-1. Solutions and Elementary Operations
Introduction to linear equations, systems, consistency, and elementary row operations.
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§ 1-2. Gaussian Elimination
Row-Echelon Form, RREF, Rank, and the Gaussian Elimination algorithm.
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§ 1-3. Homogeneous Equations
Trivial vs Nontrivial solutions, Linear Combinations, and Basic Solutions.
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§ 1-5. Application to Electrical Networks
Solving circuit problems using systems of linear equations. Kirchhoff's Laws.
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§ 1-6. Application to Chemical Reactions
Balancing chemical equations using systems of linear equations.
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§ 2-1. Matrix Operations
Addition, Scalar Multiplication, Transpose, and Symmetry.
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§ 2-2. Matrix Transformations
Vectors, Matrix-Vector Product, and Rotations in 2D.
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§ 2-3. Matrix Multiplication
Row-by-Column rule, Properties, and Non-Commutativity.
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§ 2-4. Matrix Inverses
Definition, Uniqueness, 2x2 Formula, and Algorithm.
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§ 2-5. Elementary Matrices
Three types, Equivalence to Row Operations, and Invertibility.
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§ 2-6. Linear Transformations
Definition, Matrix of T, Rotations, and Reflections.
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§ 2-7. LU Factorization
Definition, Solving Systems, and the Multiplier Method.
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§ 2-9. Markov Chains
Transition Matrices, State Vectors, and Steady States.
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Chapter 3: Determinants and Diagonalization
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§ 3-1. The Cofactor Expansion
Determinants, Cofactor Expansion, and Row Operations.
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§ 3-2. Determinants and Inverses
Product Theorem, Cramer's Rule, and Vandermonde.
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§ 3-3. Diagonalization
Eigenvalues, Eigenvectors, and the Diagonalization Theorem.
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§ 3-4. Linear Recurrences
Fibonacci Numbers and Linear Dynamical Systems.
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Chapter 4: Vector Geometry
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§ 4-1. Vectors and Lines
Norms, Unit Vectors, and Lines in Space.
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§ 4-2. Projections and Planes
Dot Product, Orthogonality, and Planes.
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§ 4-3. The Cross Product
Area, Volume, and Orthogonal Vectors.
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§ 4-4. Linear Operators on $\mathbb{R}^3$
Rotations, Reflections, and Composition.
5 Vector Space $\mathbb{R}^n$
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§ 5-1. Subspaces of $\mathbb{R}^n$
Definition, Examples, Null Space, and Image Space.
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§ 5-2. Basis and Dimension
Linear Independence, Spanning Sets, and Dimension.
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§ 5-3. Orthogonal Sets
Orthogonality, Orthonormal Sets, and Fourier Expansion.
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§ 5-4. Rank and Nullity
Row/Column Spaces, Rank Theorem, and Rank-Nullity Theorem.
6 Vector Spaces
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§ 6-1. Examples and Basic Properties
Axioms, Polynomials, Matrices, and Functions.
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§ 6-2. Subspaces and Spanning Sets
Subspace Test, Spanning Sets, and Examples.
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§ 6-3. Linear Independence and Dimension
Independence, Basis, and Dimension of General Spaces.
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§ 6-4. Finite Dimensional Spaces
Extending Basis, Reducing Spanning Sets, and Dimension Formula.
7 Linear Transformations
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§ 7-1. Examples and Elementary Properties
Definition, Matrix Transformations, and Properties.
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§ 7-2. Kernel and Image
Null Space, Range, and the Rank-Nullity Theorem.
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§ 7-3. Isomorphisms and Composition
Isomorphisms, Composition, and Inverses.
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§ 8-3. Positive Definite Matrices
Definition, Tests, and Cholesky Factorization.
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§ 8-4. QR Factorization
Gram-Schmidt in Matrix Form.
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§ 8-5. Singular Value Decomposition
The Pinnacle of Linear Algebra.