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Prash
Linear Algebra
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Prash
Our Index of Tutorials for all the topics
Mathematics
Algebra
Introduction to Complex NumbersIntroduction to Complex Numbers and iota. Argand plane and iota. Complex numbers as free vectors. Nth roots of a complex number. Notes, formulas and solved problems related to these subtopics. Series and ProgressionsArithmetic, Geometric, Harmonic and mixed progressions. Notes, formulas and solved problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic means and the relationship between them. The Principle of Mathematical Induction Introductory problems related to Mathematical Induction. Quadratic EquationsIntroducing various techniques by which quadratic equations can be solved  factorization, direct formula. Relationship between roots of a quadratic equation. Cubic and higher order equations  relationship between roots and coefficients for these. Graphs and plots of quadratic equations. Quadratic Inequalities Quadratic inequalities. Using factorization and visualization based methods. 
Geometry
Coordinate Geometry
Introduction to Coordinate Geometry The Equation of Straight Line The Circle A Quick Introduction to Conic Sections: Parabola, Hyperbola, Ellipse Parabola Hyperbola Ellipse 
Probability
Linear Algebra
Linear Algebra  Matrices Part I  A Tutorial with Examples Introduction to Matrices. Theory, definitions. What a Matrix is, order of a matrix, equality of matrices, different kind of matrices: row matrix, column matrix, square matrix, diagonal, identity and triangular matrices. Definitions of Trace, Minor, Cofactors, Adjoint, Inverse, Transpose of a matrix. Addition, subtraction, scalar multiplication, multiplication of matrices. Defining special types of matrices like Symmetric, Skew Symmetric, Idempotent, Involuntary, Nilpotent, Singular, NonSingular, Unitary matrices. Linear Algerba  Matrices Part II  A Tutorial with Examples, Problems and Solutions Problems and solved examples based on the subtopics mentioned above. Some of the problems in this part demonstrate finding the rank, inverse or characteristic equations of matrices. Representing real life problems in matrix form. Linear Algebra  Determinants  A Tutorial with Examples, Problems and Solutions Introduction to determinants. Second and third order determinants, minors and cofactors. Properties of determinants and how it remains altered or unaltered based on simple transformations is matrices. Expanding the determinant. Solved problems related to determinants. Linear Algebra  Simultaneous Equations in Multiple Variables  A Tutorial with Examples and Problems Representing a system of linear equations in multiple variables in matrix form. Using determinants to solve these systems of equations. Meaning of consistent, homogeneous and nonhomogeneous systems of equations. Theorems relating to consistency of systems of equations. Application of Cramer rule. Solved problems demonstrating how to solve linear equations using matrix and determinant related methods. Basic Concepts In Linear Algebra and Vector Spaces  A Tutorial with Examples and Solved ProblemsTheory and definitions. Closure, commutative, associative, distributive laws. Defining Vector space, subspaces, linear dependence, dimension and bias. A few introductory problems proving certain sets to be vector spaces. Linear Algebra  Introductory Problems Related to Vector SpacesProblems demonstrating the concepts introduced in the previous tutorial. Checking or proving something to be a subspace, demonstrating that something is not a subspace of something else, verifying linear independence; problems relating to dimension and basis; inverting matrices and echelon matrices. Linear Algebra  More about Vector Spaces Defining and explaining the norm of a vector, inner product, GrahamSchmidt process, coordinate vectors, linear transformation and its kernel. Introductory problems related to these. Linear Algebra  Linear Transformations, Operators and Maps Solved examples and problems related to linear transformation, linear maps and operators and other concepts discussed theoretically in the previous tutorial. Linear Algebra  Eigenvalues, Eigenvectors and Cayley Hamilton Theorem Eigenvalues, eigenvectors, Cayley Hamilton Theorem Linear Algebra  Problems Based on Simultaneous Equations, Eigenvalues, EigenvectorsDemonstrating the Crammer rule, using eigenvalue methods to solve vector space problems, verifying Cayley Hamilton Theorem, advanced problems related to systems of equations. Solving a system of differential equations . Linear Algebra  A few closing problems in Recurrence Relations Solving a recurrence relation, some more of system of equations. 
Vectors
Trigonometry
 Trigonometry 1a ( Introduction to Trigonometry  Definitions, Formulas ) Introducing trigonometric ratios, plots of trigonometric functions, compound angle formulas. Domains and ranges of trigonometric functions, monotonicity of trigonometric functions quadrant wise. Formulas for double and triple angle ratios. Trigonometry 1b ( Tutorial with solved problems based on Trigonometric ratios ) Problems based on the concepts introduced above. Trigonometry 2a ( Basic concepts related to Heights and Distances ) Applying trigonometry to problems involving heights and distances. Angles of elevation and depression. Sine and Cosine rule, half angle formulas. Circumradius, inradius and escribed radius. Circumcentre, incentre, centroid and median of a triangle. Trigonometry 2b ( Tutorial with solved problems related to Heights and Distances and other applications of Trigonometry )  Problems based on the concepts introduced above. Trigonometry 3a ( Introducing Inverse Trigonometric Ratios)  Inverse trigonometric ratios  their domains, ranges and plots. Trigonometry 3b ( Tutorial with solved problems related to inverse trigonometric ratios ) Problems related to inverse trigonometric ratios. Trigonometry 4 ( A tutorial on solving trigonometric equations ) Solving trigonometric equations. Methods and transformations frequently used in solving such equations. 
Single Variable Calculus
 Quick and introductory definitions related to Funtions, Limits and Continuity  Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits; defining the L'Hospital rule, intermediate and extreme value theorems. Functions, Limits and Continuity  Solved Problem Set I  The Domain, Range, Plots and Graphs of Functions; L'Hospital's Rule  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule. Functions, Limits and Continuity  Solved Problem Set II  Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial). Functions, Limits and Continuity  Solved Problem Set III  Continuity and Intermediate Value Theorems  Problems related to Continuity, intermediate value theorem. Introductory concepts and definitions related to Differentiation  Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc  Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem, Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. Differential Calculus  Solved Problem Set I  Common Exponential, Log , trigonometric and polynomial functions  Examples and solved problems  differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . Differential Calculus  Solved Problem Set II  Derivability and continuity of functins  Change of Indepndent Variables  Finding Nth DerivativesExamples and solved problems  related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the Nth derivative of functions. Differential Calculus  Solved Problems Set III Maximia, Minima, Extreme Values, Rolle's Theorem  Examples and solved problems  related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem. Differential Calculus  Solved Problems Set IV  Points of Inflexion, Radius of Curvature, Curve Sketching  Examples and solved problems  Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves. Differential Calculus  Solved Problems Set V  Curve Sketching, Parametric Curves  More examples of investigating and sketching curves, parametric representation of curves. Introducing Integral Calculus  Definite and Indefinite Integrals  using Substitution , Integration By Parts, ILATE rule  Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions. Definite integration and properties of definite integrals; Integration by substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. Integral Calculus  Solved Problems Set I  Basic examples of polynomials and trigonometric functions, area under curves  Examples and solved problems  elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves. Integral Calculus  Solved Problems Set II  More integrals, functions involving trigonometric and inverse trigonometric ratios  Examples and solved problems  integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. Integral Calculus  Solved Problems Set III  Reduction Formulas, Using Partial FractionsI Examples and solved problems  Reduction formulas, reducing the integrand to partial fractions, more of definite integrals. Integral Calculus  Solved Problems Set IV  More of integration using partial fractions, more complex substitutions and transformations Examples and solved problems  More of integrals involving partial fractions, more complex substitutions and transformations Integral Calculus  Solved Problems Set V Integration as a summation of a series  Examples and solved problems  More complex examples of integration, examples of integration as the limit of a summation of a series. Introduction to Differential Equations and Solved Problems  Set I  Order and Degree, Linear and NonLinear Differential Equations, Homogeneous Equations, Integrating Factor  Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). Differential Equations  Solved Problems  Set II  D operator, auxillary equation, General Solution  Examples and solved problems  Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI ) Differential Equations  Solved Problems  Set III  More Differential Equations  More complex cases of differential equations. Differential Equations  Solved Problems  Set IV  Still more differential equations. 
Multiple Variable Calculus
Applied Mathematics : An Introduction to Game Theory
 An Introduction to Game Theory Extensive Games Bayesian Games : Games with Incomplete Information Repeated Games 
Applied Mathematics : An Introduction to Operations Research
Introduction to Operations Research A quick introduction to Operations Research. Introducing Linear Programming, standard and canonical forms. Linear Programming geometry, feasible regions, feasible solutions, simplex method. Some basic problems. 
Physics
Basic Mechanics
 Introduction to Vectors and Motion Vectors and Projectile Motion Newton's Laws of Motion Work, Force and Energy Simple Harmonic Motion Rotational Dynamics Fluid Mechanics 
Engineering Mechanics
 Moments and Equivalent Systems Centroid And Center of Gravity Analysis of Structures 
Electrostatics and Electromagnetism
 Electrostatics  Part 1: Theory, definitions and problems Columb's law. Electric Field Intensity, principle of superposition, gauss theorem, electrostatic potential, electric field intensities due to common charge distributions, capacitors and calculating capacitance. Solved problems. Electrostatics  Part 2: More solved problems. More solved problems related to the concepts introduced above. Electromagnetism  Part 1: Theory and Definitions Lorentz Force, BioSavart law, Ampere's force law, basic laws related to Magnetic fields and their applications. Magnetic field intensities due to common current distributions. Electromagnetic Induction. Self and mutual induction. Electromagnetism  Part 2: Solved problems Solved problems related to the concepts introduced above. Advanced concepts in Electrostatics and Electromagnetism ( Theory only ) Advanced concepts related to electrostatics and electromagnetism (theory only). 
Computer Science and Programming
Data Structures and Algorithms
Arrays : Popular Sorting and Searching Algorithms
 Bubble Sort  One of the most elementary sorting algorithms to implement  and also very inefficient. Runs in quadratic time. A good starting point to understand sorting in general, before moving on to more advanced techniques and algorithms. A general idea of how the algorithm works and a the code for a C program. Insertion Sort  Another quadratic time sorting algorithm  an example of dynamic programming. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs insertion sort. Selection Sort  Another quadratic time sorting algorithm  an example of a greedy algorithm. An explanation and step through of how the algorithm works, as well as the source code for a C program which performs selection sort. Shell Sort An inefficient but interesting algorithm, the complexity of which is not exactly known. Merge Sort An example of a Divide and Conquer algorithm. Works in O(n log n) time. The memory complexity for this is a bit of a disadvantage. Quick Sort In the average case, this works in O(n log n) time. No additional memory overhead  so this is better than merge sort in this regard. A partition element is selected, the array is restructured such that all elements greater or less than the partition are on opposite sides of the partition. These two parts of the array are then sorted recursively. Heap Sort Efficient sorting algorithm which runs in O(n log n) time. Uses the Heap data structure. Binary Search Algorithm Commonly used algorithm used to find the position of an element in a sorted array. Runs in O(log n) time. 
Basic Data Structures and Operations on them
Stacks Last In First Out data structures ( LIFO ). Like a stack of cards from which you pick up the one on the top ( which is the last one to be placed on top of the stack ). Documentation of the various operations and the stages a stack passes through when elements are inserted or deleted. C program to help you get an idea of how a stack is implemented in code. Queues First in First Out data structure (FIFO). Like people waiting to buy tickets in a queue  the first one to stand in the queue, gets the ticket first and gets to leave the queue first. Documentation of the various operations and the stages a queue passes through as elements are inserted or deleted. C Program source code to help you get an idea of how a queue is implemented in code. Single Linked List A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind. Double Linked List A self referential data structure. A list of elements, with a head and a tail; each element points to another of its own kind in front of it, as well as another of its own kind, which happens to be behind it in the sequence. Circular Linked List Linked list with no head and tail  elements point to each other in a circular fashion. 
Tree Data Structures
Binary Search Trees A basic form of tree data structures. Inserting and deleting elements in them. Different kind of binary tree traversal algorithms. Heaps  A tree like data structure where every element is lesser (or greater) than the one above it. Heap formation, sorting using heaps in O(n log n) time. Height Balanced Trees  Ensuring that trees remain balanced to optimize complexity of operations which are performed on them. 
Graphs and Graph Algorithms
 Depth First Search  Traversing through a graph using Depth First Search in which unvisited neighbors of the current vertex are pushed into a stack and visited in that order. Breadth First Search  Traversing through a graph using Breadth First Search in which unvisited neighbors of the current vertex are pushed into a queue and then visited in that order. Minimum Spanning Trees: Kruskal Algorithm Finding the Minimum Spanning Tree using the Kruskal Algorithm which is a greedy technique. Introducing the concept of Union Find. Minumum Spanning Trees: Prim's Algorithm Finding the Minimum Spanning Tree using the Prim's Algorithm. Dijkstra Algorithm for Shortest Paths Popular algorithm for finding shortest paths : Dijkstra Algorithm. Floyd Warshall Algorithm for Shortest Paths All the all shortest path algorithm: Floyd Warshall Algorithm Bellman Ford Algorithm  Another common shortest path algorithm : Bellman Ford Algorithm. 
Popular Algorithms in Dynamic Programming
Dynamic Programming A technique used to solve optimization problems, based on identifying and solving subparts of a problem first. Integer Knapsack problemAn elementary problem, often used to introduce the concept of dynamic programming. Matrix Chain Multiplication Given a long chain of matrices of various sizes, how do you parenthesize them for the purpose of multiplication  how do you chose which ones to start multiplying first? Longest Common Subsequence Given two strings, find the longest common sub sequence between them. Dynamic Programming Algorithms covered previously: Insertion Sort, Floyd Warshall Algorithm Algorithms which we already covered, which are example of dynamic programming. 
Greedy Algorithms
Elementary cases : Fractional Knapsack Problem, Task Scheduling  Elementary problems in Greedy algorithms  Fractional Knapsack, Task Scheduling. Along with C Program source code. Data Compression using Huffman TreesCompression using Huffman Trees. A greedy technique for encoding information. 
Commonly Asked Programming Interview Questions  from Microsoft/Google/Facebook/Amazon interviews
Programming Interview Questions with Solutions  Microsoft, Google, Facebook, Amazon 
A Collection of C Programs
Functional Programming Principles and Techniques
Functional Programming  A General Overview  Using the Functional Programming paradigm with a regular programming language like Ruby  
Introduction to Ruby
 Introduction to Ruby and some playing around with the Interactive Ruby Shell (irb) Introduction to Ruby  Conditional statements and Modifiers: Ifthen, Unless, Case Introduction to Ruby Comments  Single and MultiLine comments Introduction to Ruby Loops  Using While, Until, For, Break, Next , Redo, Retry Introduction to Ruby  Arrays  Sorting, Filtering (Select), Transforming, MultiDimensional Arrays Introduction to Ruby  Strings Introduction to Ruby  Making a Script Executable Introduction to Ruby  Regular Expressions, Match, Scan Introduction to Ruby  Computing Factorials Recursively : An Example of Recursion Introduction to Ruby  Binomial Coefficients (nCr) : An Example of Recursion Introduction to Ruby  Computing a Power Set : An Example of Recursion Introduction to Ruby  Towers of Hanoi : An Example of Recursion Introduction to Ruby  Strings: Substitution, Encoding, BuiltIn Methods 
Basic Data Structures & Collections With Ruby
 Basic Data Structures in Ruby  Insertion Sort Basic Data Structures in Ruby  Selection Sort Basic Data Structures in Ruby  Merge Sort Basic Data Structures in Ruby  Quick Sort Functional Programming with Ruby Basic Data Structures in Ruby  Stack Basic Data Structures in Ruby  The Queue Basic Data Structures in Ruby  Linked List  ( A Simple, Singly Linked List) Basic Data Structures in Ruby  Binary Search Tree 
Databases  A Quick Introduction To SQL  Sample Queries demonstrating common commands
Introduction to SQL A few sample queries  A Case Study  Coming up with a Schema for Tables Taking a look at how the schema for a database table is defined, how different fields require to be defined. Starting with a simple "case study" on which the following SQL tutorials will be based.  Introduction to SQL A few sample queries : Creating Tables (CREATE) Creating tables, defining the type and size of the fields that go into it.  Introduction to SQL  A few sample queries : Making Select Queries Elementary database queries  using the select statement, adding conditions and clauses to it to retrieve information stored in a database.  Introduction to SQL  A few sample queries : Insert, Delete, Update, Drop, Truncate, Alter Operation Example of SQL commands which are commonly used to modify database tables. 
Introduction to SQL  A few sample queries: Important operators  Like, Distinct, Inequality, Union, Null, Join, Top Other Important SQL operators.  Introduction to SQL A few sample queries: Aggregate Functions  Sum, Max, Min, Avg  Aggregate functions to extract numerical features about the data. 
Introduction To Networking
Client Server Program in Python  A basic introduction to networking and client server programming in Python. In this, you will see the code for an expression calculator . Clients can sent expressions to a server, the server will evaluate those expressions and send the output back to the client. 
Introduction to Basic Digital Image Processing Filters
Introductory Digital Image Processing filters  Lowpass/Blurring filters, hipass filters and their behavior, edge detection filters in Matlab . You can take a look at how different filters transform images. Matlab scripts for these filters. 
An Introduction to Graphics and Solid Modelling
Electrical Science and Engineering
Introduction to DC Circuits
 Circuit Theory 1a Introduction to Electrical Engineering, DC Circuits, Resistance and Capacitance, Kirchoff Law Resistors, Capacitors, problems related to these. Circuit Theory 1b  More solved problems related to DC Circuits with Resistance and Capacitance Capacitors, computing capacitance, RC Circuits, time constant of decay, computing voltage and electrostatic energy across a capacitance Circuit Theory 2a  Introducing Inductors Inductors, inductance, computing selfinductance, fluxlinkages, computing energy stored as a magnetic field in a coil, mutual inductance, dot convention, introduction to RL Circuits and decay of an inductor. Circuit Theory 2b  Problems related to RL, LC, RLC circuits Introducing the concept of oscillations. Solving problems related to RL, LC and RLC circuits using calculus based techniques. Circuit Theory 3a  Electrical Networks and Network Theorems Different kind of network elements: Active and passive, linear and nonlinear, lumped and distributed. Voltage and current sources. Superposition theorem, Thevenin (or Helmholtz) theorem and problems based on these. Circuit Theory 3b  More network theorems, solved problems More solved problems and examples related to electrical networks. Star and Delta network transformations, maximum power transfer theorem, Compensation theorem and Tellegen's Theorem and examples related to these. 
Introduction to Digital Electronic Circuits and Boolean logic
 Introduction to the Number System : Part 1 Introducing number systems. Representation of numbers in Decimal, Binary,Octal and Hexadecimal forms. Conversion from one form to the other. Number System : Part 2 Binary addition, subtraction and multiplication. Booth's multiplication algorithm. Unsigned and signed numbers. Introduction to Boolean Algebra : Part 1Binary logic: True and false. Logical operators like OR, NOT, AND. Constructing truth tables. Basic postulates of Boolean Algebra. Logical addition, multiplication and complement rules. Principles of duality. Basic theorems of boolean algebra: idempotence, involution, complementary, commutative, associative, distributive and absorption laws. Boolean Algebra : Part 2Demorgan's laws. Logic gates. 2 input and 3 input gates. XOR, XNOR gates. Universality of NAND and NOR gates. Realization of Boolean expressions using NAND and NOR. Replacing gates in a boolean circuit with NAND and NOR. Understanding Karnaugh Maps : Part 1 Introducing Karnaugh Maps. Minterms and Maxterms. Canonical expressions. Sum of products and product of sums forms. Shorthand notations. Expanding expressions in SOP and POS Forms ( Sum of products and Product of sums ). Minimizing boolean expressions via Algebraic methods or map based reduction techniques. Pair, quad and octet in the context of Karnaugh Maps. Karnaugh Maps : Part 2 Map rolling. Overlapping and redundant groups. Examples of reducing expressions via KMap techniques. Introduction to Combinational Circuits : Part 1 Combinational circuits: for which logic is entirely dependent of inputs and nothing else. Introduction to Multiplexers, Demultiplexers, encoders and decoders.Memories: RAM and ROM. Different kinds of ROM  Masked ROM, programmable ROM.
Combinational Circuits : Part 2 Static and Dynamic RAM, Memory organization.
Introduction to Sequential Circuits : Part 1 Introduction to Sequential circuits. Different kinds of Flip Flops. RS, D, T, JK. Structure of flip flops. Switching example. Counters and Timers. Ripple and Synchronous Counters.
Sequential Circuits : Part 2ADC or DAC Converters and conversion processes. Flash Converters, ramp generators. Successive approximation and quantization errors. 
Introduction to Electrical Energy Generation and Power Electrical Engineering
Wind Energy  A Tutorial  Wind, Wind Machines, Wind Energy Conversions, Wind Turbines  Components and Configurations
A Tutorial on Power Generation From Thermal Power Plants  TS Graph, Process, Principles, Turbine Model, Basic Boiler Schematics
Other Renewable Sources of Energy  Solar Energy, Biomass
Economics of Electrical Power Generation
Estimation of Capacitance and Inductance in Electrical Transmission Lines
Insulation of Electrical Cables
An Introduction to Power System Protection
Introduction to Electrical Energy Generation and Power Electrical Engineering
