Linear Algebra

Lecture 1 - Introduction to the Course Contents

Lecture 2 - Linear Equations

Lecture 3a - Equivalent Systems of Linear Equations I : Inverses of Elementary Row-operations, Row-equivalent matrices

Lecture 3b - Equivalent Systems of Linear Equations II : Homogeneous Equations, Examples

Lecture 4 - Row-reduced Echelon Matrices

Lecture 5 - Row-reduced Echelon Matrices and Non-homogeneous Equations

Lecture 6 - Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations

Lecture 7 - Invertible matrices, Homogeneous Equations Non-homogeneous Equations

Lecture 8 - Vector spaces

Lecture 9 - Elementary Properties in Vector Spaces. Subspaces

Lecture 10 - Subspaces (Continued...), Spanning Sets, Linear Independence, Dependence

Lecture 11 - Basis for a vector space

Lecture 12 - Dimension of a vector space

Lecture 13 - Dimensions of Sums of Subspaces

Lecture 14 - Linear Transformations

Lecture 15 - The Null Space and the Range Space of a Linear Transformation

Lecture 16 - The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces

Lecture 17 - Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank - I

Lecture 18 - Equality of the Row-rank and the Column-rank - II

Lecture 19 - The Matrix of a Linear Transformation

Lecture 20 - Matrix for the Composition and the Inverse. Similarity Transformation

Lecture 21 - Linear Functionals. The Dual Space. Dual Basis - I

Lecture 22 - Dual Basis II. Subspace Annihilators - I

Lecture 23 - Subspace Annihilators - II

Lecture 24 - The Double Dual. The Double Annihilator

Lecture 25 - The Transpose of a Linear Transformation. Matrices of a Linear Transformation and its Transpose

Lecture 26 - Eigenvalues and Eigenvectors of Linear Operators

Lecture 27 - Diagonalization of Linear Operators. A Characterization

Lecture 28 - The Minimal Polynomial

Lecture 29 - The Cayley-Hamilton Theorem

Lecture 30 - Invariant Subspaces

Lecture 31 - Triangulability, Diagonalization in Terms of the Minimal Polynomial

Lecture 32 - Independent Subspaces and Projection Operators

Lecture 33 - Direct Sum Decompositions and Projection Operators - I

Lecture 34 - Direct Sum Decompositions and Projection Operators - II

Lecture 35 - The Primary Decomposition Theorem and Jordan Decomposition

Lecture 36 - Cyclic Subspaces and Annihilators

Lecture 37 - The Cyclic Decomposition Theorem - I

Lecture 38 - The Cyclic Decomposition Theorem - II. The Rational Form

Lecture 39 - Inner Product Spaces

Lecture 40 - Norms on Vector spaces. The Gram-Schmidt Procedure I

Lecture 41 - The Gram-Schmidt Procedure II. The QR Decomposition

Lecture 42 - Bessel's Inequality, Parseval's Indentity, Best Approximation

Lecture 43 - Best Approximation: Least Squares Solutions

Lecture 44 - Orthogonal Complementary Subspaces, Orthogonal Projections

Lecture 45 - Projection Theorem. Linear Functionals

Lecture 46 - The Adjoint Operator

Lecture 47 - Properties of the Adjoint Operation. Inner Product Space Isomorphism

Lecture 48 - Unitary Operators

Lecture 49 - Unitary operators - II. Self-Adjoint Operators - I.

Lecture 50 - Self-Adjoint Operators - II - Spectral Theorem

Lecture 51 - Normal Operators - Spectral Theorem