MAL 101 (Mathematics II-27W)

Syllabus (Pg 167) 

There will be 42 lectures of 55 minutes each.

Office Hours: Appointment basis.

Evaluation Plan

• End-Semester Examination: 35 marks

• Mid-Semester Examination: 25 marks

• Quizzes: 2 quizzes, 15 marks each (total 30 marks)

• Presentation: 10 marks (Please refer to the instructions provided under the tentative course plan)

Tentative course plan is the following. A reference code of the form [Xm.n] refers to the subsection m.n of the reference identified by [X] in the references section below.

• Lecture 01 : Equivalent system of linear quations have exactly the same solutions. [H1.2]

• Lecture 02 : Elementary row operations and their inverse; row equivalent matrices and solutions of homogeneneous system. [H1.2]

• Lecture 03 : Row-reduced (RR) matrix and how to obtain them from any given matrix, identity matrix, Kronecker delta. [H1.3]

• Lecture 04 : Row-reduced echelon (RRE) matrix and how to obtain them from any given matrix, impartance of such matrices in solving homogeneous systems of linear equations. [H1.4]

• Lecture 05 : Gauss elimination to transform any matrix into an upper triangular matrix, Augmented matrix. [H1.4].

Alternatively, students may refer to [B3.1].

• Lecture 06 : Process of applying the elementary row operations may be seen using the familiar process of multiplication of matrices,  matrix multiplication is non-commutative and associative. [H1.5]

• Lecture 07 : Elementary matrices and its applications. [H1.5]

• Lecture 08 : Invertability and elementary matrices. [H1.6]

• Lecture 09 : LU-decomposition, existence and uniqueness. [H1.5] (Theorem 9 of [H1.5] alng with its corollary and Theorem 5 gives this idea.)

Alternatively, students may refer to [B3.5].

• Lecture 10 : Vector spaces with examples and properties. [F1.1, 1.2]

• Lecture 11 : Subspaces with examples; intersections, union ans sums of subspaces. [F1.3]

• Lecture 12 : linear combination, span. [F1.4]

• Lecture 13 : linear dependence and independence, bases. [F1.5]

• Lecture 14 : Replacement theorem (Lagrange interpolation), the idea of dimension. [F1.6]

• Lecture 15 : Existence of basis. [F1.7]

Quiz 1 (February)

• Lecture 16 : Range space and rank, Null space and nullity, Rank nullity theorem. [F2.1]

• Lecture 17 : Linear transformation (LT) and their existence with examples. (L(V,W) and its dimension, algebra of linear transformations). [F2.1]

• Lecture 18 : Isomorphism, Matrix representation of LT. [F2.2] 

• Lecture 19 : Linear independence, rank of a matrix. [F 3.2]

• Lecture 20 : Solutions of linear systems, existence and uniqueness. [F3.3]

• Lecture 21 : Diagonalizability of an LT, eigenvalues and eigenvectors. [F5.1]

• Lecture 22 : The characteristic polynomial. [F5.1]

• Lecture 23 : Diagonalization. [F5.2]

Mid-semester examination (March)

• Lecture 24 : Direct sums. [F5.2]

• Lecture 25 : Cayley-Hamilton theorem. [F5.4]

• Lecture 26 : Inner product spaces with examples. [F6.1]

• Lecture 27 : Gram-Schmidt process. [F6.2] 

• Lecture 28 : Hermitian matrices, orthonormal bases, unitary matrices. [F6.3-6.4]

• Lecture 29 : Real quadratic form. [F6.5]

• Lecture 30 : Minimal polynomial and diagonalizability. [F7.3]

• Lecture 31 : Review of First Order ODE. Separable and homogeneous. [K1.1-1.3]

• Lecture 32 : Exact, integrating factor, Homogeneous and non-homogeneous linear ODEs, Bernoulli equation. [K1.4-1.5]

• Lecture 33 : Existence and uniquenesss, Lipschitz condition, Picard`s iteration method. [K1.7]

• Lecture 34 : Homogeneous linear ODE and basis of solutions. [K2.1, 2.2]

• Lecture 35 : Cauchy-Euler equations. [K2.5]

Quiz 2 (April)

• Lecture 36 : Linear dependence and Wronskian characteristic equations. [K2.6]

• Lecture 37 : Method of undetermined coefficients. [K2.7]

• Lecture 38 : Method of variation of parameters. [K2.10]

• Lecture 39 : Linear differential equations of Higher order (analogous statements, without proof). [K3.1-3.3]

• Lecture 40 : Laplace Transform. [K6.1-6.2]

• Lecture 41 : Solutions using Laplace Transform. [K6.3-6.4]

• Lecture 42 : Solutions using Laplace Transform. [K6.5-6.6]

End-semester examination (May)

Instructions for Presentations

Groups of four/five students will be assigned a course-related mathematical topic. Each group is expected to study the topic and identify a relevant application, preferably related to the engineering disciplines of its members.

Each group will give a 10-12 minute board presentation covering:

• the underlying mathematical idea and its essential concepts; and

• a relevant application, explaining how the mathematical idea is used in that application.

Evaluation will focus on the group's understanding of the mathematics, understanding of the application, and ability to clearly explain the connection between the two. Merely describing an application without demonstrating an understanding of the underlying mathematics will not be sufficient.

Presentations (board/slides) will be conducted in my office, unless an alternative arrangement is agreed upon in advance. Groups are encouraged to use examples, illustrations, or simple computations where appropriate.

Preparation and Scheduling

Groups will be given seven days to prepare after receiving their topic. They may be asked to present on any day after the seventh day, with one day's notice. For example, if a topic is assigned on February 1, the group may be asked to present on any day from February 9 onward.

Originality of Presentation

Groups assigned the same topic are expected to develop their presentations independently. If a group's presentation is substantially identical to that of a group that presented earlier, the group will receive 2 marks less than the earlier group's score. Groups assigned the same topic are therefore advised to discuss the topic with one another, while developing their own applications, explanations, and presentations independently.

References

• [F] Friedberg, Stephen H., Insel, Arnold J., & Spence, Lawrence E. (2003). Linear Algebra (4th ed.). Prentice-Hall of India Pvt. Ltd. New Delhi.

• [H] Hoffman, Kenneth, & Kunze, Ray Alden (1978). Linear Algebra (2nd ed.). Prentice Hall of India Pvt. Limited. Delhi. Pearson Education India Reprint, 2015.

• [K] E. Kreyszig (2007), Advanced Engineering Mathematics (9th edition) John Wiley.

• [B] (Optional) Bradie, Brian. (2006). A Friendly Introduction to Numerical Analysis. Pearson Education India. Dorling Kindersley (India) Pvt. Ltd. Third impression 2011.