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.

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

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.

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