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