- What's a linear space?
- Linear Spaces: Examples and non-Examples
- Some facts about linear spaces.
- Subspaces and Linear Combinations.
- Introduction: Dependent and Independent Sets.
- Basic Facts about Dependent and Independent Sets.
- Bases, Dimension, and Components.
- Distance and Geometry in Linear Spaces
- The Cauchy-Schwarz-Buniakovsky Inequality.
- Orthogonality in Euclidean Spaces.
- Parseval's Theorem and Pythagoras's Theorem.
- Orthogonalization/Gram Schmidt
- Orthogonal Completents S-Perp
- Projections
- An Application to Data Analysis
- Linear Transformations
- Examples of Linear Transformations
- Null Space, Range, and Kernel
- Examples
- An Important Theorem about Dimensions
- Algebra of Linear Transformations
- Inverses
- One to One, Invertible and Consequences
- Extending a Linear Transformation - Bases
- Matrices
- Examples
- Diagonalization
- Algebra of Matrices
- Relationship between Linear Transformations and Matrices
- Matrix Multiplication
- Systems of Linear Equations
- Computation Techniques
- Inverting Matrices
- Basic Computational Facts for 2 and 3 Dimensional Determinants
- Motivation and Axioms for Determinants
- Uniqueness
- Computational Ideas
- Product Formulae and Inverse Matrices
- Independence of Vectors -- Block Matrices
- Expansion Formulae, Minors and Cofactors
- Existence of Determinants - Another View
- Transpose
- Cofactors and Cramer's Rule
- Eigenvalues
- Eigenvectors
- Transformations in the Plane
- Diagonalizing Matrices
- The Characteristic Polynomial
- More Eigenvalues and Eigenvectors
- Change of Basis and Similar Matrices
- The Inner Product
- Transformations and the Inner Product
- Hermitian Matrices and Orthonormal Bases
- More about Hermitian Matrices
- Unitary Matrices
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