A basic Text Book of Linear Algebra

A book with applications

A book with lots of Examples

Lectures:

**Tuesday 8:30 am (LT-2) Thursday 8:30 am (LT-2) and****Friday 8:30 am (LT-2)****First Class on July 25, 2013 !****Note: No Class on July 23, 2013 due to Orientation Program****Instructor:****Manish K Gupta (****www.mankg.com****)****Office: Room 2209 Faculty Block 2****Email:****mankg@daiict.ac.in****Phone: 91-79-30510****549****Attendance Policy:****Each of you must attend each lecture as I usually give a few questions (called type-2 questions) that you need to solve by that week itself and clear your doubts about it. Note that I usually ask them in the exams or quizzes. There could be a surprise quiz at any time in Lectures and sometime I may change the % of quiz for final grade to quite a lot and may include its % for grade as well.****If for some reason beyond your control (for example you are sick) you are about to miss a lecture please send an email in advance to me (subject line: your roll number, name and SC 116 Fall 2013) that you will not be able to attend the lecture that day.****Marks Distribution (Tentative) / Grading Policy:**Test 1

20%

Test 2

20%

Test 3

30%

Quiz/Tutorials/Class

20%

Take Home

10%

**Tutorials Policy:****One Tutorial per week in two parts****Part-1 (Theoretical)****Part-2 (Symbolic Computing: Matlab, Mathematica or Maple etc.)****You need to submit your Tutorial every week and attendance is mandatory****Monday (Group 2, 14:00 to 14:55 and Group 1, 15:00 to 15:55), Room CEP 203****Thursday (****Group 4, 14:00 to 14:55 and Group 3, 15:00 to 15:55), Room CEP 203****Tutor: Group 2 & Group 1, Prof. Puneet Bhateja****Tutor: Group 4 & Group 3, Prof.****Puneet Bhateja****Tutor Office: Faculty Block No. 2, Room No 2101, Ground Floor****TA1: Group 2 & Group 1, Krishna Gopal****TA2: Group 4 & Group 3, Sarita Agrawal****TA3: Group 1, 2, 3 & 4, Snehal Bhayani (Matlab Help)**Linear Algebra Wiki

Watch YouTube Video (24 minutes Video)

## Weekly outline

### 22 July - 28 July

Introduction & History

Tutorial 1

The Birth of Algebra video (1:44:23) Professor Keith Devlin

Solutions 1

Q1 matlab

Q2 matlab

Q3 matlab

Q4 matlab

### 29 July - 4 August

Number System, Fields, Systems of Linear Equations, Matrices and Elementary Row Operations, Row Reduced Echelon Form, Operations on Matrices (A+B, AB, A^{-1}…), Some Properties

Tutorial 2

Solutions 2

### 5 August - 11 August

Vector Spaces: R^n, C^n, Subspaces, Dual Space, linear independence (l.i.), Span, Bases, Dimension, Coordinates

Tutorial 3

Solutions 3

### 12 August - 18 August

Linear Transformations and its Algebra, Isomorphism, Image and Kernal of LT

Tutorial 4

### 19 August - 25 August

Representation by Matrices, Linear Functionals, Polynomial Space

**Revision Summary ( Aug 22 ) and Test 1 (Aug 23 - Aug 27)**Please note it is a sample test and only posted for style of the exam. The questions could be quite different in difficulty level and otherwise. Use it for your practice, fun and time management. Last problem will be a star problem to find stars in the class. There are 4 questions sampled from the topics coevred before the Test 1.

Enjoy !

Quiz Week

Test 1 Solutions

### 26 August - 1 September

Inner Product Spaces (IPS), Norm Linear Spaces (NLS): Examples

Tutorial 5

Marks for Test 1 SC 116 have been posted in Lecture Folder:

Max Marks = 40 (only one student)

Avg. = 11.23

Min. =0

Solutions Tutorial 5

### 2 September - 8 September

Cauchy-Schwarz Inequality, Angle between two vectors, Orthogonal Complement, Fundamental Theorem of Linear Algebra, Orthonormal Sets and Basis,

Gram-Schmidt Orthogonalization Process, Orthogonal Projections and QR decomposition

Tutorial 6

Solutions 6

### 9 September - 15 September

Eigenvalues and Eigenvectors, Characteristic Polynomial, Cayley-Hamilton Theorem, Diagonalization of Matrices, Special Matrices, Unitary Equivalence,

Spectral Theorem for Normal Matrices, Bilinear Form, SVD

Tutorial 7

**Solutions 7 posted in Lecture folder (9 MB )**

### 16 September - 22 September

The_Prisoner_of_Benda

Group Theory, Examples of finite and infinite group, Subgroups, Cyclic Groups, Symmetric Group:S_n

Plus : Groups

Tutorial 8

Solutions Tutorial 8

### 23 September - 29 September

Order of an Element and Group, Lagrange's Theorem, Fermat Theorem, Cosets, Normal Subgroups, Factor Group, Homomorphism, Isomorphism,

Fundamental Theorem of Isomorphism, Simple Group.

### 30 September - 6 October

**Revision Summary (Oct 1)****Test 2 (Oct 3, 2013, 11:00-13:00 hrs)****Duration 2 hrs****Total 4 Questions (pattern is similar to Test 1)**Revise

Test 2 Solutions

### 7 October - 13 October

Cauchy's Theorem for Abelian Groups, Sylow's Theorem for Abelian Groups, Automorphisms, Cayley's Theorem, Permutation Groups, Counting Principle,

Sylow's Theorem

Tutorial 9

Solutions 9

### 14 October - 20 October

Direct Products, Finite Abelian Groups, Defenition and Examples of Rings, Some Special Classes of Rings, Homomorphosims

Tutorial 10

Tutorial Solutions 10

### 21 October - 27 October

Ideals and Factor Rings, Maximal Ideals, Factor (Quotient) Rings, The Field of Quotients of an Integral Domain, Euclidean Rings

Tutorial 11

Tutorial Solutions 11

### 28 October - 3 November

Ring of Gaussian Integers, Polynomial Rings, Polynomials over the Rational Field, Polynomials Rings over Commutative Rings

### 4 November - 10 November

**Semester Break (Nov 4, 2013 - November 10, 2013)****Take Home Assignment**Formulate a real life application problem into a mathematical problem & solve it using your knowledge of SC116 course.

Hint: The problem may be using concepts from linear algebra, groups, rings or fields etc.

Note: You are free to take problems from your other courses or internet. But direct (without understanding) copy-paste is not allowed. The assignment has to be submitted individually (not in groups). Try to be different.

The solution should be typed using LaTex. A .pdf and corresponding .tex files (both) should be uploaded.

A .tex sample file for that is available (see files below for application files .tex, .bib etc)

The file should have following sections:

1. Describe the application /motivation

2. Formulate the Mathematics

3. Solve the Mathematics

4. Interpretation and significance of the solution for application

5. Bibliography

The name of the file should be your rollno.tex. For example, it could be

201226354.tex and 201226354.pdf and 201226354.bib

If you have image file you can upload that as well with name 201226354_fig.jpg

Please do not zip the files and submit seprate files.

The best way to prepare Bibliography is using bibtex. Learn how you can use it.

Latex Help on Windows Machine:

Step 1: Install Ghostscript & GSview (http://pages.cs.wisc.edu/~ghost/)

/>Step 2: Install Adobe Reader (http://get.adobe.com/reader/) You can skip it if you already have it on your system

Step 2: Install Miktex (http://miktex.org/) : You can install basic one or full

step 3: Install Latex Editor (http://www.texniccenter.org/)

/>

In step 3, it will ask for path for miktex bin file just specify that and you are ready to use it.

Info:

1. You may require latex math symbols list (http://www.ctan.org/tex-archive/info/symbols/comprehensive/symbols-a4.pdf)

/>2. Get basic info about latex at wiki (http://en.wikipedia.org/wiki/LaTeX) you can download some basic book if you want.

3. One thing that you may want to learn is how to include figures or graphics in tex document see (ftp://ftp.tex.ac.uk/tex-archive/info/epslatex.pdf)

/>4. You can draw figures in Xfig (http://www.xfig.org/)

/>

See some samples in lecture folder for applications but not all of them are good.

Good luck & Happy Deepwali !

ManishSmaple PDF File

Sample Project Latex File. You may modify this.

Sample Bibliography File. You may modify this.

### 11 November - 17 November

**Course Evaluation (Nov 14, 2013 - Nov 21, 2013)**Finite Fields and Finite Rings, Extension Fields, Automorphisms and Fields, Splitting Fields

Tutorial 12

Tutorial Solutions 12

### 18 November - 24 November

Applications of Algebraic Structures: Data Storage, Image Processing, Signal Processing, Communication, Coding Theory, Security and Cryptography, VLSI etc.

### 25 November - 1 December

**Test 3 (November 25, 2013 - November 30, 2013)**Test 3 (November 30, 2013, 14:00-16:00 hrs) Duration 2 hrs Total 4 Questions (pattern is similar to Test 1)

CEP 102, 110 and 108