Linear Algebra

Description

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