# ISL Linear Algebra (2014)

 Venue: CEMS, SSJ Campus, Kumaun University, Almora Dates: 17th – 29th March, 2014

 School Convener(s) Name D. P. Patil H.S.Dhami Sanjay K Pant Mailing Address Department of Mathematics, Indian Institute of Science, Bangalore-560012 Vice-chancellor, Kumaun University,  Nainital 263001 Deen Dayal Upadhyaya College, University of Delhi, Karampura, New Delhi-110015

 Speakers and Syllabus

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,
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

 Date 9.30-11.00 11.30-13.00 14.30-15.30 16.00-17.00 17/3/14 JKV SAK T1 T2 18/3/14 JKV SAK T3 T4 19/3/14 JKV SAK T5 T6 20/3/14 JKV SAK T7 T8 21/3/14 JKV SAK T9 T10 22/3/14 JKV SAK T11 T12 24/3/14 SRG DPP T13 T14 25/3/14 SRG DPP T15 T16 26/3/14 SRG DPP T17 T18 27/3/14 SRG DPP T19 T20 28/3/14 SRG DPP T21 T22 29/3/14 SRG DPP T23 T24

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

 Selected Applicants

FINAL LIST
(Go through the list carefully and then write to me (if you have any querry))

Very -Very Important:

• GUEST HOUSE ACCOMMODATION IS ON SHARING BASIS. NO ACCOMPONYING GUEST WILL BE ALLOWED. WE REPEAT AGAIN NO GUEST WILL BE ALLOWED.
• PLEASE DO NOT EMBARRASS US BY MAKING A REQUEST FOR GUEST ACCOMMODATION.UNIVERSITY GUEST HOUSE HAS LIMITED CAPACITY.

Those who needs selection/invitation letter for leave from their institute/college/university should send mail to sanjpant@gmail.com . Please do not forget to mention followings:

1. Title (Dr./Mr/Ms)
2. Name
3. Department
4. Address of institution where you are working with PIN-Code.

Remarks

1. In March-April Almora and surroundings are at their BEST.
***It is moderately cold***
*** It is season of flowers***
2. In case of any querry please feel free to contact Sanjay Kumar Pant :
Mobile: 919810528236 E-mail: sanjpant@gmail.com
3. For updates visit : www.cemsalmora.org

*** If anytime you decide that you will not come for the school , Please let us know.***

How to reach

• Travel By Air: The nearest airport is New Delhi. From Delhi one can reach Almora by train or by road as given below.
•  By Train: The nearest train station is Kathgodam. Several important rail stations in India have direct connection to Kathgodam. From Delhi, there three main trains run almost daily from Delhi to Kathgodam: 12040 ANVT-KGM Shatabdi (06:15 AM), 15013 Ranikhet Express (22:40) and 15035 UTR-SAMPARK Express (16:00). We recommend the first two connections. One may also get down at Haldwani (the station just before Kathgodam). Please note that train number 12040 starts from ANAND VIHAR railway station at Delhi and the other two trains depart from OLD DELHI railway station. So, if you plan to reach Kathgodam by train, after arriving at the airport or New Delhi railway station, you should direct yourself towards the Anand Vihar railway terminus or the Old Delhi rail station in Delhi. The metro stops most nearest to these rail stations are Anand Vihar and Chandni Chowk respectively.

Summary of trains:
*1. Delhi - Kathgodam by Ranikhet Express. (Train No. 15013)

*2. Anand Vihar (ANVT)-Kathgodam (KGM) Shatabdi (Train o. 12040)

3. Delhi - Kathgodam by Uttarakhand Smprk Kranti Express (Train No.15035)

4. Delhi - Lal Kuan by Pooja Express. (Train No. 0517)

Trains with asterix are recommended.

•  By Road: This option is little expensive, but direct and in many cases, would be flexible. This option is to travel to Almora from New Delhi Airport by road. You can book a (private) taxi at the airport or in advance. By car it generally takes 8-10 hours to Almora (from Delhi airport) assuming that you will take one or two breaks in between the journey.

AC Volvo buses are also available from ISBT Delhi to Kathgodam.

• Car Direction: Delhi - Hapur bypass - Moradabad bypass - Rampur - Bilaspur - Rudrapur - Haldwani - Kathgodam [approximately 270 kms]- Ranibagh - Bhimtal - Bhowali - Khairna Bridge - Kwarah Bridge - Almora. From Kathgodam/Haldwani to Almora: One has to travel this route by car or bus. It takes sround 3 hrs to reach Almora from Kathgodam. We recommend you to hire a car or shared taxi at Kathgodam/Haldwani. If you intend to travel by a shared taxi, it is better to get down at Haldwani since the availability of shared taxis are better there. On the way to Almora from Kathgodam/Haldwani, you will have to cross some beautiful tourist locations, for example, Bheemtal (http://www.bheemtal.com/nainit/bheemtal.html).