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Indian Institute of Science Education and Research Bhopal

Mathematics

MTH 311: Advanced Linear Algebra (4)

Learning Objectives:

This course reviews undergraduate linear algebra and proceeds to more advanced topics. Its purpose is to provide a solid understanding of linear algebra of the sort needed throughout graduate mathematics.

Course Contents:

Vector spaces, subspaces, bases and dimension, some examples.

Linear transformations, rank - nullity theorem, the algebra of linear transformations, invertible linear transformations, matrix of a linear transformation, change of basis, linear functionals, annihilator of a subspace, dual space, double dual, canonical isomorphism between a vector space and its double dual, transpose of a linear transformation.

Characteristic values, diagonalizable linear operator, equivalent notions of diagonalizable operator (in terms of characteristic polynomial, dimensions of eigen spaces), annihilating polynomials, minimal polynomial, characterization of diagonalizable operator using the minimal polynomial, invariant subspaces, simultaneous triangulation, simultaneous diagonalization, direct sum decompositions, projections, invariant direct sums, primary decomposition theorem, nilpotent operators, S-N decomposition.

Rational and Jordan forms.

Inner product space, Gram-Schmidt, linear functionals and adjoints, unitary operators, normal operators, self-adjoint operators, spectral theorem for self-adjoint operators.

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