Linear Algebra I

Overview

Introduction to the Course Contents - Linear Equations - Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations - Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples - Row-reduced Echelon Matrices - Row-reduced Echelon Matrices and Non-homogeneous Equations - Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations - Invertible matrices, Homogeneous Equations Non-homogeneous Equations - Elementary Properties in Vector Spaces. Subspaces - Subspaces (continued), Spanning Sets, Linear Independence, Dependence - Basis for a vector space - Dimension of a vector space - Dimensions of Sums of Subspaces - Linear Transformations - The Null Space and the Range Space of a Linear Transformation - The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces - Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank - The Matrix of a Linear Transformation - Matrix for the Composition and the Inverse. Similarity Transformation - Linear Functionals - The Dual Space - Dual Basis - Subspace Annihilators - Subspace Annihilators

The Transpose of a Linear Transformation. Matrices of a Linear - The Double Dual. The Double Annihilator - Eigenvalues and Eigenvectors of Linear Operators - Diagonalization of Linear Operators. A Characterization - The Minimal Polynomial - The Cayley-Hamilton Theorem - Invariant Subspaces - Triangulability, Diagonalization in Terms of the Minimal Polynomial - Independent Subspaces and Projection Operators - Direct Sum Decompositions and Projection Operators - The Primary Decomposition Theorem and Jordan Decomposition - Cyclic Subspaces and Annihilators - The Cyclic Decomposition Theorem - The Rational Form - Inner Product Spaces - Norms on Vector spaces - The Gram-Schmidt Procedure - The QR Decomposition - Bessel's Inequality, Parseval's Indentity, Best Approximation - Best Approximation: Least Squares Solutions - Orthogonal Complementary Subspaces, Orthogonal Projections - Projection Theorem. Linear Functionals - The Adjoint Operator - Properties of the Adjoint Operation. Inner Product Space Isomorphism - Unitary Operators - Self-Adjoint Operators - Spectral Theorem