# TEW - Linear Algebra and Calculus on Rn

## Speakers and Syllabus

This Teachers’ Enrichment Workshop is especially meant for mathematics teachers of arts and science colleges in and around Chennai. The goal of this workshop to introduce and indicate the interplay between linear algebra and calculus of several variables. Then we intend to introduce basics of bilinear forms,especially over real numbers. Finally, we cover inverse and implicit function theorems finishing with the rank theorem which are foundational to numerous branches of mathematics. The workshop will be an intensive one and only those who are willing to work hard and actively participate are encouraged to apply.

 Speaker Affiliation Topic(s) Number of lecture hours Number of tutorial hours Detailed Syllabus S. Gun IMSc Bilinear forms 5 4 Bilinear forms, Matrix representations; Quadratic forms, Discriminants; Quadratic forms over complex numbers; Quadratic forms over real numbers (Sylvester’ law). A. Mukhopadhyay IMSc Introduction to Calculus on Rn 6.5 3 Differentiable maps, Basic Properties; Smooth functions, Diffeomorphisms , Examples; Partial Derivatives and Chain Rule; Equality of mixed partial derivatives; Jacobian Matrix, Mean Value Theorem. P. Rath CMI Inverse and Implicit function Theorems 6.5 3 Taylor’s formula; Inverse function Theorem; Implicit function Theorem; Rank theorem; Introduction to Bump functions. P. Sankaran IMSc Vector spaces and Linear Transformations 6 2 Vector spaces, Bases; Linear transformations; Eigen values and eigen vectors; Diagonalisation and Triangularisation of linear operators; Determinants of linear transformations and their elementary properties.

Reference books :

• Spivak : Calculus on Manifolds;
• J. Taylor : Foundations of Analysis;
• Herstein : Topics in Algebra;
• Hoffman and Kunze : Linear algebra.

## Time Table

 Time 09:45–11:15 11:30 to 13:00 14:00 to 15:00 15:15 to 16:15 Mon 20th PS L1 AM L1 L AM L2 AM/PS T1 S Tue 21st PS L2 T AM L3 U SG L1 T SG/PS T2 N Wed 22nd PS L3 E AM L4 N SG L2 E SG/AM T3 A Thu 23rd PS L4 A PR L1 C AM L5 A AM/PR T4 C Fri 24th PR L2 SG L3 H PR L3 SG/PR T5 K Sat 25th PR L4 SG L4 PR L5 SG/PR T6 S

SG = S. Gun, AM = A. Mukhopadhyay, PR= P. Rath, PS= P. Sankaran
Ln = Lecture number n, Tn = Tutorial number n

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