ISL Linear Algebra (2014)

Module I (Vector Spaces)
Lecture I-1(Vector Spaces):Vector spaces, Subspaces, Linear System of equations, Gauss Elimination.
Lecture I-2(Bases and Dimension): Generating systems, Bases and Dimensions of vector spaces.
Lecture I-3(Linear Maps): Linear Maps, Space of Linear maps, Linear maps and bases.
Lecture I-4(Affine Spaces): Affine spaces and affine subspaces, Affine maps, reflections,rotations, Affine groups, Description of affine groups in dimension 2 and 3.
Lecture I-5(The rank theorem): The rank theorem, Direct sums and projections.
Lecture I-6(Dual Spaces): Dual spaces, dual bases, dual of a linear map, hyperplanes, Quotient spaces, codimension, isomorphism theorems, Group actions.
Instructor : Prof. J.K. Verma(JKV), Department of Mathematics, IIT Bombay-400076 (jkv at math.iitb.ac.in)
Tutor : TBA

Module II (Matrices and Determinants)
Lecture II-1(Matrices): The matrix of a linear map, relation to change of bases, inverties, transpose, Rank of matrices, row rank, column rank.
Lecture II-2(Elementary Matrices): Elementary matrices, Elementary operations.
Lecture II-3(Determinants): Permutations, Canonical decomposition, Signature, Symmetric and Alternative groups, Multi-linear maps, Determinant functions.
Lecture II-4(Computational Rules): Computational rules for Determinants, Expansion of determinants in terms of rows and columns,
Adjoint formula, Cramer’s rule.
Lecture II-5(Dual Spaces): Determinant of a linear map.
Lecture II-6(Determinants and Volumes): Orientations and Volumes.
Instructor : Prof. S.A. Katre(SAK), Department of Mathematics, University of Pune, Pune-400007(sakatre at gmail.com)
Tutor : TBA

Module III (Linear Operators)
Lecture III-1(Polynomial algebras): Polynomials over a field, Division with remainder, Prime polynomials, Euclid’s lemma, Uniqueness of prime decomposition, Zeroes of polynomials, Prime polynomials in Q[X], R[X] and in C[X].Eisenstein irreducibility criterion.
Lecture III-2(Characteristic polynomial and minimal polynomial): Eigen-values, Spectrum, Characteristic polynomial, Cayley Hamilton theorem, Companion matrices, Invariant subspaces.
Lecture III-3(Diagonalizable and Triangulable operators): Eigenspaces, Algebraic and geometric multiplicities, Characterization of diagonalizable and triangulable operators.
Lecture III-4(Some decomposition Theorems): Primary decomposition theorem, Additive and multiplicative canonical decompositions.
Lecture III-5(The Jordan Normal forms): Jordan Normal forms
Lecture III-6(Applications): System of linear differential equations with constant coefficients.
Instructor : Prof. S.R. Ghorpade(SRG), Department of Mathematics, IIT Bombay-400076 (srg at math.iitb.ac.in)
Tutor: TBA

Module IV (Sesqui-linear forms)
Lecture IV-1(Bilinear and Sesqui-linear forms):Sesqui-linear functions, Gram’s matrix and Gram; criterion.
Lecture IV-2(Duality and Orthogonalisation): Non-degeneracy and complete duality, Symmetric and Complex Hermitian forms, Schmidt’s orthogonalisation. .
Lecture IV-3(Types of hermitian forms): Sylvester’s Law of inertia and Hurwitz’s criterion.
Lecture IV-4(Vector spaces with scalar products): Scalar products, Cauchy-Schwartz inequality, Minkowski’s inequality,Orthogonal
projections, Bassel’s inequality, Vector products, Volumes in Euclidean spaces, Volume of the paralleotops.
Lecture IV-5(Isometries): Linear isometry and affine isometries, Rigid motions, especially of R2 and R3 .
Lecture IV-6(The Spectral Theorems): Self-adjoint and Normal operators, Principal axis theorem.
Instructor : Prof. D.P. Patil(DPP), Department of Mathematics, Indian Institute of Science,Bangalore-560012(patil at maths.iisc.ernet.in)
Tutor: TBA

Texts/References
1. Artin,M.:Algebra, Prentice Hall, 1994.
2. Halmos,P.R.:Finite-Dimensional Vector Spaces, Springer-Verlag, 1993.
3. Herstein,I.N.:Topics in Algebra, Wiley Eastern, 1987.
4. Hoffman,K. and Kunze,R.:Linear Algebra, Prentice-Hall, 1972.
5. Jacobson,N.:Basic Algebra, Vols I and II, Hindustan Pub.Co.,1984.
6. Greub,W.:Linear Algebra, Springer-Verlag, GTM 97,(4th ed.), 1981.

Tentative Time-table

JKV—J.K.Verma
SAK—S.A.Katre
SRG—S.R.Ghorpade
DPP—D.P.Patil
Tn— nth tutorial
Tea Breaks—11.00-11.30 and 15.30-16.00
Lunch Break— 13.00 - 14.30

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