Linear Algebra

Linear Systems
	Solving Linear Systems
		Gauss\' Method
		Describing the Solution Set
		General = Particular + Homogeneous
	Linear Geometry of n-Space
		Vectors in Space
		Length and Angle Measures*
	Reduced Echelon Form
		Gauss-Jordan Reduction
		Row Equivalence
	Topic: Computer Algebra Systems
	Topic: Input-Output Analysis
	Topic: Accuracy of Computations
	Topic: Analyzing Networks
Vector Spaces
	Definition of Vector Space
		Definition and Examples
		Subspaces and Spanning Sets
	Linear Independence
		Definition and Examples
	Basis and Dimension
		Basis
		Dimension
		Vector Spaces and Linear Systems
		Combining Subspaces*
	Topic: Fields
	Topic: Crystals
	Topic: Voting Paradoxes
	Topic: Dimensional Analysis
Maps Between Spaces
	Isomorphisms
		Definition and Examples
		Dimension Characterizes Isomorphism
	Homomorphisms
		Definition
		Rangespace and Nullspace
	Computing Linear Maps
		Representing Linear Maps with Matrices
		Any Matrix Represents a Linear Map*
	Matrix Operations
		Sums and Scalar Products
		Matrix Multiplication
		Mechanics of Matrix Multiplication
		Inverses
	Change of Basis
		Changing Representations of Vectors
		Changing Map Representations
	Projection
		Orthogonal Projection Into a Line*
		Gram-Schmidt Orthogonalization*
		Projection Into a Subspace*
	Topic: Line of Best Fit
	Topic: Geometry of Linear Maps
	Topic: Markov Chains
	Topic: Orthonormal Matrices
Determinants
	Definition
		Exploration*
		Properties of Determinants
		The Permutation Expansion
		Determinants Exist*
	Geometry of Determinants
		Determinants as Size Functions
	Other Formulas
		Laplace\'s Expansion*
	Topic: Cramer\'s Rule
	Topic: Speed of Calculating Determinants
	Topic: Projective Geometry
Similarity
	Complex Vector Spaces
		Factoring and Complex Numbers; A Review*
		Complex Representations
	Similarity
		Definition and Examples
		Diagonalizability
		Eigenvalues and Eigenvectors
	Nilpotence
		Self-Composition*
		Strings*
	Jordan Form
		Polynomials of Maps and Matrices*
		Jordan Canonical Form*
	Topic: Method of Powers
	Topic: Stable Populations
	Topic: Linear Recurrences
Appendix
	Propositions
		Quantifiers
		Techniques of Proof
		Sets, Functions, and Relations