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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.

 Lectures Topic 1 Introduction to the Course Contents. 2 Linear Equations 3a Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations, Row-equivalent matrices 3b Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples 4 Row-reduced Echelon Matrices 5 Row-reduced Echelon Matrices and Non-homogeneous Equations 6 Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations 7 Invertible matrices, Homogeneous Equations Non-homogeneous Equations 8 Vector spaces 9 Elementary Properties in Vector Spaces. Subspaces 10 Subspaces (continued), Spanning Sets, Linear Independence, Dependence 11 Basis for a vector space 12 Dimension of a vector space 13 Dimensions of Sums of Subspaces 14 Linear Transformations 15 The Null Space and the Range Space of a Linear Transformation 16 The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces 17 Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I. 18 Equality of the Row-rank and the Column-rank II 19 The Matrix of a Linear Transformation 20 Matrix for the Composition and the Inverse. Similarity Transformation 21 Linear Functionals. The Dual Space. Dual Basis I 22 Dual Basis II. Subspace Annihilators I 23 Subspace Annihilators II 24 The Double Dual. The Double Annihilator 25 The Transpose of a Linear Transformation. Matrices of a Linear Transformation and its Transpose 26 Eigenvalues and Eigenvectors of Linear Operators 27 Diagonalization of Linear Operators. A Characterization 28 The Minimal Polynomial 29 The Cayley-Hamilton Theorem 30 Invariant Subspaces 31 Triangulability, Diagonalization in Terms of the Minimal Polynomial 32 Independent Subspaces and Projection Operators 33 Direct Sum Decompositions and Projection Operators I 34 Direct Sum Decomposition and Projection Operators II 35 The Primary Decomposition Theorem and Jordan Decomposition 36 Cyclic Subspaces and Annihilators 37 The Cyclic Decomposition Theorem I 38 The Cyclic Decomposition Theorem II. The Rational Form 39 Inner Product Spaces 40 Norms on Vector spaces. The Gram-Schmidt Procedure I 41 The Gram-Schmidt Procedure II. The QR Decomposition 42 Bessel's Inequality, Parseval's Indentity, Best Approximation 43 Best Approximation: Least Squares Solutions 44 Orthogonal Complementary Subspaces, Orthogonal Projections 45 Projection Theorem. Linear Functionals 46 The Adjoint Operator 47 Properties of the Adjoint Operation. Inner Product Space Isomorphism 48 Unitary Operators 49 Unitary operators II. Self-Adjoint Operators I 50 Self-Adjoint Operators II - Spectral Theorem 51 Normal Operators - Spectral Theorem
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