University Dept. of Mathematics, Dr. Shyama Prasad Mukherji University (DSPMU), Ranchi

A 4-day TIMC-Workshop on **‘Linear Algebra and Applications’** was organized in the University Dept. of Mathematics, Dr. Shyama Prasad Mukherji University (DSPMU), Ranchi during 10-13, 2019 under the aegis of ‘The Indian Mathematics Consortium’ (TIMC). Jharkhand Society of Mathematical Sciences (JSMS), a Society member of TIMC got DSPMU coordinated with the Consortium for this workshop. Prof. S. A. Katre, Chair Professor, Lokmanya Tilak Chair, S. P. Pune University and Treasurer, TIMC was the convener of this workshop. Besides himself, he associated Prof. K. C. Prasad, Retired Prof., R.U., Dr. Ganesh Kadu, Asst. Prof., S.P. Pune University and Dr. S. Paul, Asst. Prof., RKMV University, Belur (W.B.) as three additional resource persons for the workshop.

Four classes each of 75 minutes duration along with one 1-hour guest lecture in semi-technical/expository vein was organized daily. In guest lectures students were acquainted with:

(i) relevance of mathematics and sister subjects in emerging science, technology and A. I., as well as with focused applications of linear algebra in

(ii) solving a system of linear differential equations,

(iii) state of population of genotypes-a topic in Genetics and

(iv) stability of dynamical systems of 2-body and 3- body problems

by Prof. D. K. Singh, Director, BIT, Sindri; Prof. B. K. Rajhans, Retd. Prof., IIT (ISM), Dhanbad; Prof. M. K. Singh, Retd. Prof., R. U. and Dr. Govind Jha, Vinoba Bhave University, H’bag respectively. Also on 12^{th}, one long (4.30 to 6 pm) doubt clearing tutorial session was engaged by Mr. Himanshu Pandey (recorder of the Workshop) in the lead role .

Given that students had a familiarity with the subject through usual course work, it was decided to tune the students in the essential thought process emphasizing on natural course of evolution of the ideas through examples and abstraction in line with ‘Linear Algebra and Applications’ by David C Lay et al. Analysing what the students have already been doing in the name of solving a system of linear equations right since school days, and common Indian practice of record keeping of properties in tabular form, Prof. K. C. Prasad digged out some essential ideas and building blocks of linear algebra namely: row operations, reduced echelon forms, consistency, rank of a table, algebra of n-tuples , algebra of matrices and by abstraction that of vector spaces, role of a matrix as a linear transformation, elementary matrices, factorization of a nonsingular matrix as a product of elementary matrices etc.. Dr. S. Paul discussed properties of a vector space, subspace, span, notion of linear independence, basis and dimension. Dr. Kadu discussed linear transformations from a vector space to itself and associated invariant subspaces, change of basis theorem and similar matrices. He found matrices of rotation, projection and reflection in R^{2 }and R^{3} and some related invariant subspaces, eigenvalues, eigenvectors, eigenspace, algebraic and geometric multiplicity of an eigenvalue and diagonalisability of a square matrix. Devoting more on applications, Prof. Katre worked out all bases of (Rn, R, +, ( ) )and examined V_{n+1 }= {all polynomials over R of degree ≤ n} U {0} . All bases of V_{n+1 } were obtained using the standard basis. It was discussed that Lagrange polynomials used in Lagrange interpolation formula form a basis of this space. Prof. Katre also shew that the set of all Bernstein polynomials of degree n forms a basis of V_{n+1 }and used it to indicate a constructive proof of Weierstrass approximation theorem. A classical problem of design engineers has been to find suitable curves associated with local controls such that movement of one control point would affect a small part of the curve only. Katre discussed Bezier’s solution of this problem which adds extra control points wisely and associates a class of special curves named ‘Bezier curves’ for the needful. Construction of Bezier surfaces used in drawing models of cars was discussed. He finally enlightened the students on use of linear algebra in Google search for ranking of webpages.

80 students (20 from senior Honours classes and 60 from PG classes) of different colleges and Universities (DSPMU, RU, CUJ) of Ranchi attended this workshop. On the first two days attendance was cent percent. On third and fourth days we found 4 and 8 absentees respectively. Breakfast, two tea and lunch were arranged for the participants of the workshop in UGC HRDC Centre (Staff College) canteen of Ranchi University at 5 minutes walking distance from the venue of the workshop. Resource persons and guests stayed in HRDC guest house. Participants were called for breakfast at 8 AM. Opening time of the Workshop was 9 AM, Lunch time 1PM to 2 PM and closing time for tea- at- venue was 5 PM modulo marginal fluctuations of timing on account of 1 hour inaugural and valedictory sessions.

The inaugural function (10.8.2019 : 9 AM to 10 AM) was chaired by Prof. N. C. Das, President, JSMS. Retd. Professor of Mathematics IIT (ISM) Dhanbad was the chief guest and Sri Shailesh Kumar Singh, IAS, and Principal Secretary to the Department of Higher and Technical Education, GOJ was the guest of honour. Dr. A. K. Jha anchored the program. Organising Secretary, Dr. Ashoke Kumar Mahato welcomed the guests. Key note addresses were delivered by Prof. S. A. Katre and Prof. K. C. Prasad on behalf of TIMC and JSMS respectively. After a very delighting and informative lecture of the Principal Secretary about U- tube lectures of Professors of Standford University and other centres of excellence on topics of concern and suggestions of the chief guest on how to optimize the benefits of the workshop by the participants, Prof A. K. Choudhary, Director, HRDC, RU, Dr. N. D. Goswami, Registrar DSPMU and Mr. B. K. Singh, Director, School of Pharmacy, Ranchi delivered good wishes to the participants and organizers of the workshop and ensured all possible assistance. The function closed after brief Presidential address on Mathematics inspired moral values and vote of thanks by Dr. R. A. Tiwari Ex HOD, RU.

The valedictory function (13.8.2019: 4.30 to 5.30 PM) was chaired by Prof. S. N. Munda,

V. C., DSPM University, Ranchi. Surgery of the program including suggestions were delivered by the convenor Dr. S. A. Katre. Students’ feedback, distribution of certificates and address of the President including his recommendation for more such workshops responding to students’ narration of benefits of it, were the essential features of the valedictory function.