Course Archives Theoretical Statistics and Mathematics Unit
Course: Linear Algebra
Level: Postgraduate
Time: Currently not offered
Past Exams


i) Quick review of solutions of a system of linear equations, vector spaces, sub- spaces, linear independence and span, Zorns lemma and existence of basis, quotient spaces and direct sum of vector spaces, exact sequences and splittings, linear maps and matrices, matrix of a linear map in a basis, invertibility, rank and determinant, linear functionals, dual space, annihilator, transpose of a linear map.
[This part is mostly a review and should be covered quickly with emphasis on problem solving.]
ii) Eigenvalues, algebraic and geometric multiplicities, characteristic and minimal polynomials, upper triangularization, diagonalizability and semisimplicity, decomposition into nilpotent and semisimple matrices, Cayley-Hamilton Theorem.
iii) Tensor product of vector spaces, extension of scalars, complexification, tensor product of linear maps, symmetric and exterior algebra, determinant as a multilinear map and Laplace expansion.
iv) Inner-product spaces, orthogonality, Gram-Schmidt orthogonalization, Bessels inequality, projection and orthogonal projection, symmetric and Hermitian operators, orthogonal and unitary diagonalizability, normal operators, spectral the- orem, bilinear and quadratic forms, positive definite operator, square-root of a positive operator, polar decomposition, isometry, rigid motions, the rotation group.
v) Structure theory of finitely generated modules over PID and application to canonical forms.

Suggested Texts :
(a) D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley (Asian reprint 2003).
(b) S. Lang, Algebra, GTM (211), Springer (Indian reprint 2002).
(c) K. Hoffman and R. Kunze, Linear Algebra, Prentice-Hall of India (1998).
(d) N.S. Gopalakrishnan, University Algebra, Wiley Eastern (1986).
(e) A. R. Rao and P. Bhimasankaram, Linear Algebra, TRIM(19), Hindustan Book Agency (2000).
(f) P. R. Halmos, Finite-Dimensional Vector Spaces: Second Edition.

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Past Exams
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