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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.
Systems of linear equations, equivalent systems, matrices
Elementary row operations, properties, examples
Row-reduced echelon matrices, elementary matrices,
Invertible matrices, elementary matrices, equivalence
Other equivalent conditions in terms of systems of linear equations, homogeneous, non-homogeneous, examples
Vector spaces, definition and examples
Linear independence and dependence, examples
Basis, dimension, examples
Ordered basis, examples, coordinates
Linear transformations, examples
Range space, null space, examples
Injective, Surjective linear transformations, examples
Rank-nullity theorem, consequences, examples
Algebra of linear transformations
Matrix representation of linear transformations, examples
Change of bases, matrix representation, examples
Linear functionals, annihilator
Double dual, examples
Dual basis, properties, examples
Transpose of a linear transformation, properties
Characteristic Values and characteristic vectors, examples
Properties of characteristic values, vectors, examples
Similarity transformation, Diagonalizability, characterization
Minimal polynomial, properties and relationship with the characteristic polynomial
Cayley-Hamilton theorem, applications
Invariant subspaces, characteristic values, examples
Direct-sum decomposition, examples, properties
Invariant direct-sum decomposition, examples
The Primary decomposition Theorem I
The Primary decomposition Theorem II
Cyclic subspaces, annihilators, examples
Cyclic decomposition Theorem I
Cyclic decomposition Theorem II
Rational form, examples
Jordan form, examples
Inner product spaces, examples
Cauchy-Schwarz inequality, examples
Orthonormal bases, Gram-Schmidt procedure.
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.