Mathematics - Linear Algebra (English)

   Saturday, May 03, 2025   

Starting a new Lecture Notes Series on Mathematics - Linear Algebra

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Lecture 1: Mod-01 Lec-01 Introduction to the Course Contents.

Lecture 2: Mod-01 Lec-02 Linear Equations

Lecture 3: Mod-01 Lec-03a Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations

Lecture 4: Mod-01 Lec-03b Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples

Lecture 5: Mod-01 Lec-04 Row-reduced Echelon Matrices

Lecture 6: Mod-01 Lec-05 Row-reduced Echelon Matrices and Non-homogeneous Equations

Lecture 7: Mod-01 Lec-06 Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations

Lecture 8: Mod-01 Lec-07 Invertible matrices, Homogeneous Equations Non-homogeneous Equations

Lecture 9: Mod-02 Lec-08 Vector spaces

Lecture 10: Mod-02 Lec-09 Elementary Properties in Vector Spaces. Subspaces

Lecture 11: Mod-02 Lec-10 Subspaces (continued), Spanning Sets, Linear Independence, Dependence

Lecture 12: Mod-03 Lec-11 Basis for a vector space

Lecture 13: Mod-03 Lec-12 Dimension of a vector space

Lecture 14: Mod-03 Lec-13 Dimensions of Sums of Subspaces

Lecture 15: Mod-04 Lec-14 Linear Transformations

Lecture 16: Mod-04 Lec-15 The Null Space and the Range Space of a Linear Transformation

Lecture 17: Mod-04 Lec-16 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces

Lecture 18: Mod-04 Lec-17 Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I

Lecture 19: Mod-04 Lec-18 Equality of the Row-rank and the Column-rank II

Lecture 20: Mod-05 Lec19 The Matrix of a Linear Transformation

Lecture 21: Mod-05 Lec-20 Matrix for the Composition and the Inverse. Similarity Transformation

Lecture 22: Mod-06 Lec-21 Linear Functionals. The Dual Space. Dual Basis I

Lecture 23: Mod-06 Lec-22 Dual Basis II. Subspace Annihilators I

Lecture 24: Mod-06 Lec-23 Subspace Annihilators II

Lecture 25: Mod-06 Lec-24 The Double Dual. The Double Annihilator

Lecture 26: Mod-06 Lec-25 The Transpose of a Linear Transformation. Matrices of a Linear

Lecture 27: Mod-07 Lec-26 Eigenvalues and Eigenvectors of Linear Operators

Lecture 28: Mod-07 Lec-27 Diagonalization of Linear Operators. A Characterization

Lecture 29: Mod-07 Lec-28 The Minimal Polynomial

Lecture 30: Mod-07 Lec-29 The Cayley-Hamilton Theorem

Lecture 31: Mod-08 Lec-30 Invariant Subspaces

Lecture 32: Mod-08 Lec-31 Triangulability, Diagonalization in Terms of the Minimal Polynomial

Lecture 33: Mod-08 Lec-32 Independent Subspaces and Projection Operators

Lecture 34: Mod-09 Lec-33 Direct Sum Decompositions and Projection Operators I

Lecture 35: Mod-09 Lec-34 Direct Sum Decomposition and Projection Operators II

Lecture 36: Mod-10 Lec-35 The Primary Decomposition Theorem and Jordan Decomposition

Lecture 37: Mod-10 Lec-36 Cyclic Subspaces and Annihilators

Lecture 38: Mod-10 Lec-37 The Cyclic Decomposition Theorem I

Lecture 39: Mod-10 Lec-38 The Cyclic Decomposition Theorem II. The Rational Form

Lecture 40: Mod-11 Lec-39 Inner Product Spaces

Lecture 41: Mod-11 Lec-40 Norms on Vector spaces. The Gram-Schmidt Procedure I

Lecture 42: Mod-11 Lec-41 The Gram-Schmidt Procedure II. The QR Decomposition.

Lecture 43: Mod-11 Lec-42 Bessel's Inequality, Parseval's Indentity, Best Approximation

Lecture 44: Mod-12 Lec-43 Best Approximation: Least Squares Solutions

Lecture 45: Mod-12 Lec-44 Orthogonal Complementary Subspaces, Orthogonal Projections

Lecture 46: Mod-12 Lec-45 Projection Theorem. Linear Functionals

Lecture 47: Mod-13 Lec-46 The Adjoint Operator

Lecture 48: Mod-13 Lec-47 Properties of the Adjoint Operation. Inner Product Space Isomorphism

Lecture 49: Mod-14 Lec-48 Unitary Operators

Lecture 50: Mod-14 Lec-49 Unitary operators II. Self-Adjoint Operators I.

Lecture 51: Mod-14 Lec-50 Self-Adjoint Operators II - Spectral Theorem

Lecture 52: Mod-14 Lec-51 Normal Operators - Spectral Theorem