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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.
Introduction to the Course Contents.
Equivalent Systems of Linear Equations I: Inverses of Elementary Row-operations, Row-equivalent matrices
Equivalent Systems of Linear Equations II: Homogeneous Equations, Examples
Row-reduced Echelon Matrices
Row-reduced Echelon Matrices and Non-homogeneous Equations
Elementary Matrices, Homogeneous Equations and Non-homogeneous Equations
Invertible matrices, Homogeneous Equations Non-homogeneous Equations
Elementary Properties in Vector Spaces. Subspaces
Subspaces (continued), Spanning Sets, Linear Independence, Dependence
Basis for a vector space
Dimension of a vector space
Dimensions of Sums of Subspaces
The Null Space and the Range Space of a Linear Transformation
The Rank-Nullity-Dimension Theorem. Isomorphisms Between Vector Spaces
Isomorphic Vector Spaces, Equality of the Row-rank and the Column-rank I.
Equality of the Row-rank and the Column-rank II
The Matrix of a Linear Transformation
Matrix for the Composition and the Inverse. Similarity Transformation
Linear Functionals. The Dual Space. Dual Basis I
Dual Basis II. Subspace Annihilators I
Subspace Annihilators II
The Double Dual. The Double Annihilator
The Transpose of a Linear Transformation. Matrices of a Linear Transformation and its Transpose
Eigenvalues and Eigenvectors of Linear Operators
Diagonalization of Linear Operators. A Characterization
The Minimal Polynomial
The Cayley-Hamilton Theorem
Triangulability, Diagonalization in Terms of the Minimal Polynomial
Independent Subspaces and Projection Operators
Direct Sum Decompositions and Projection Operators I
Direct Sum Decomposition and Projection Operators II
The Primary Decomposition Theorem and Jordan Decomposition
Cyclic Subspaces and Annihilators
The Cyclic Decomposition Theorem I
The Cyclic Decomposition Theorem II. The Rational Form
Inner Product Spaces
Norms on Vector spaces. The Gram-Schmidt Procedure I
The Gram-Schmidt Procedure II. The QR Decomposition
Bessel's Inequality, Parseval's Indentity, Best Approximation
Best Approximation: Least Squares Solutions
Orthogonal Complementary Subspaces, Orthogonal Projections
Projection Theorem. Linear Functionals
The Adjoint Operator
Properties of the Adjoint Operation. Inner Product Space Isomorphism
Unitary operators II. Self-Adjoint Operators I
Self-Adjoint Operators II - Spectral Theorem
Normal Operators - Spectral Theorem
K.Hoffman and R. Kunze, Linear Algebra, 2nd Edition, Prentice- Hall of India, 2005.
M. Artin, Algebra, Prentice-Hall of India, 2005.
S. Axler, Linear Algebra Done Right, 2nd Edition, John-Wiley, 1999.
S. Lang, Linear Algebra, Springer UTM, 1997.
S. Kumaresan, Linear Algebra: A Geometric Approach, Prentice-Hall of India, 2004.