COURSE OUTLINE: In this course, you will learn systems of linear equations, Matrices, Elementary row operations, Row-reduced echelon matrices. Vector spaces, Subspaces, Bases and dimension, Ordered bases and coordinates. Linear transformations, Rank-nullity theorem, Algebra of linear transformations, Isomorphism, Matrix representation, Linear functionals, Annihilator, Double dual, Transpose of a linear transformation. Characteristic values and characteristic vectors of linear transformations, Diagonalizability, Minimal polynomial of a linear transformation, Cayley-Hamilton theorem, Invariant subspaces, Direct-sum decompositions, Invariant direct sums, The primary decomposition theorem, Cyclic subspaces and annihilators, Cyclic decomposition, Rational, Jordan forms. Inner product spaces, Orthonormal bases, Gram-Schmidt process.
Mod-01 Lec-01 Introduction to the Course Contents.. Mod-01 Lec-02 Linear Equations. Mod-01 Lec-03a Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations. Mod-01 Lec-03b Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples. Mod-01 Lec-04 Row-reduced Echelon Matrices. Mod-01 Lec-05 Row-reduced Echelon Matrices and Non-homogeneous Equations. Mod-01 Lec-06 Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations. Mod-01 Lec-07 Invertible matrices, Homogeneous Equations Non-homogeneous Equations. Mod-02 Lec-08 Vector spaces. Mod-02 Lec-09 Elementary Properties in Vector Spaces. Subspaces. Mod-02 Lec-10 Subspaces (continued), Spanning Sets, Linear Independence, Dependence. Mod-03 Lec-11 Basis for a vector space. Mod-03 Lec-12 Dimension of a vector space. Mod-03 Lec-13 Dimensions of Sums of Subspaces. Mod-04 Lec-14 Linear Transformations. Mod-04 Lec-15 The Null Space and the Range Space of a Linear Transformation. Mod-04 Lec-16 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces. Mod-04 Lec-17 Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I. Mod-04 Lec-18 Equality of the Row-rank and the Column-rank II. Mod-05 Lec19 The Matrix of a Linear Transformation. Mod-05 Lec-20 Matrix for the Composition and the Inverse. Similarity Transformation. Mod-06 Lec-21 Linear Functionals. The Dual Space. Dual Basis I. Mod-06 Lec-22 Dual Basis II. Subspace Annihilators I. Mod-06 Lec-23 Subspace Annihilators II. Mod-06 Lec-24 The Double Dual. The Double Annihilator. Mod-06 Lec-25 The Transpose of a Linear Transformation. Matrices of a Linear. Mod-07 Lec-26 Eigenvalues and Eigenvectors of Linear Operators. Mod-07 Lec-27 Diagonalization of Linear Operators. A Characterization. Mod-07 Lec-28 The Minimal Polynomial. Mod-07 Lec-29 The Cayley-Hamilton Theorem. Mod-08 Lec-30 Invariant Subspaces. Mod-08 Lec-31 Triangulability, Diagonalization in Terms of the Minimal Polynomial. Mod-08 Lec-32 Independent Subspaces and Projection Operators. Mod-09 Lec-33 Direct Sum Decompositions and Projection Operators I. Mod-09 Lec-34 Direct Sum Decomposition and Projection Operators II. Mod-10 Lec-35 The Primary Decomposition Theorem and Jordan Decomposition. Mod-10 Lec-36 Cyclic Subspaces and Annihilators. Mod-10 Lec-37 The Cyclic Decomposition Theorem I. Mod-10 Lec-38 The Cyclic Decomposition Theorem II. The Rational Form. Mod-11 Lec-39 Inner Product Spaces. Mod-11 Lec-40 Norms on Vector spaces. The Gram-Schmidt Procedure I. Mod-11 Lec-41 The Gram-Schmidt Procedure II. The QR Decomposition.. Mod-11 Lec-42 Bessel's Inequality, Parseval's Indentity, Best Approximation. Mod-12 Lec-43 Best Approximation: Least Squares Solutions. Mod-12 Lec-44 Orthogonal Complementary Subspaces, Orthogonal Projections. Mod-12 Lec-45 Projection Theorem. Linear Functionals. Mod-13 Lec-46 The Adjoint Operator. Mod-13 Lec-47 Properties of the Adjoint Operation. Inner Product Space Isomorphism. Mod-14 Lec-48 Unitary Operators. Mod-14 Lec-49 Unitary operators II. Self-Adjoint Operators I.. Mod-14 Lec-50 Self-Adjoint Operators II - Spectral Theorem. Mod-14 Lec-51 Normal Operators - Spectral Theorem.