MAL 101 (Mathematics II)

Syllabus (Pg 168) 

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.

• Lecture 01 : Equivalent system of linear quations have exactly the same solutions.

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

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

• 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

• Lecture 05 : Augmented matrix, Gauss elemenation to transform any matrix into its RRE form. 

• Lecture 06 : Matrix multiplication and its non-commutativity and associativity

• Lecture 07 : Elementary matrices and its applications,

• Lecture 08 : Invertability and elementary matrices

• Lecture 09 : LU-factorization, 

• Lecture 10 : Linear independence, rank of a matrix

• Lecture 11 : solutions of linear systems, existence and uniqueness. 

• Lecture 12 : Vector spaces with examples and properties. [F1.1, 1.2]

• Lecture 13 : Subspaces with examples; intersections, union ans sums of subspaces. [F1.3]

• Lecture 14 : linear combination, span. [F1.4]

• Lecture 15 : linear dependence and independence, bases. [F1.5]

Quiz 1 (February)

• Lecture 16 : Replacement theorem (Lagrange interpolation), the idea of dimension. [F1.6]

• Lecture 17 : Existence of basis. [F1.7]

• Lecture 18 : Linear transformation and their existence with examples. (L(V,W) and its dimension, algebra of linear transformations). [F2.1] 

• Lecture 19 : Range space and rank, Null space and nullity, Rank nullity theorem. [F2.1]

• Lecture 20 : Isomorphism, Matrix representation of linear transformation. [F2.2] 

• Lecture 21 : Eigenvalues and eigenvectors Eigne values-eigenvectors and some applications of eigenvalue problems, 

• Lecture 22 : the characteristic polynomial, Diagonalization, Hermitian, skew-Hermitian,

• Lecture 23 : Inner product spaces

Mid-semester examination (March)

• Lecture 24 : orthonormal bases

• Lecture 25 : unitary matrices and their eigenvalues-eigen bases

• Lecture 26 : 

• Lecture 27 : the minimal polynomial

• Lecture 28 : Gram-Schmidt process

• Lecture 29 : real quadratic form

• Lecture 30 : Cayley-Hamilton theorem

• Lecture 31 : 

• Lecture 32 : Review of First Order ODE

• Lecture 33 : Lipschitz condition

• Lecture 34 : Picard`s theorem; 

• Lecture 35 : Linear differential equations: 

Quiz 2 (April)

• Lecture 36 : Linear dependence and Wronskian

• Lecture 37 : linear ODE with constant coefficients of higher ordercharacteristic equations

• Lecture 38 : Cauchy-Euler equations

• Lecture 39 : method of undetermined coefficients

• Lecture 40 : method of variation of parameters

• Lecture 41 : Laplace Transform. 

• Lecture 42 : Solutions using Laplace Transform. 

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.