Matrix, Determinants and Inverse
Aims and objectives
In this lesson, we have discussed the representation of data using matrices and the representation of expressions using determinant. Matrix inversion, a convenient way to solve system of linear equations, is also discussed in this lesson.
After reading this lesson, you should able to understand
- Matrix and its usage in data representation.
- Determinant and the various methods of computing it.
- Cofactors and Adjoint of a matrix.
- How to find matrix Inverse? And to solve linear equations using inverse.
- Properties of determinant and invertible matrices.
- Applications of Matrix inverses in real-time simulations
Introduction
- J. Sylvester coined the term “matrix” in 1848. A matrix is a rectangular table of
elements. Matrices are used to describe linear equations, keep track of the coefficients of linear transformation and to record data that depend on multiple parameters. The determinant notation is now employed in almost every……Read More
Eigenvaules and Eigenvectors
Aims and objectives
In this lesson, we have discussed about the Linear Transformation, Eigenvalues and Eigenvectors.
After reading this lesson, you should able to
- Perform Linear Transformations.
- Find eigenvalues and eigenvectors.
- Know about the properties of eigenvalues and eigenvectors.
- Understand Cayley-Hamilton theorem and to find the inverse of a matrix.
- To solve problems using the properties of the eigenvalues and eigenvectors.
Introduction
Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).
The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the……Read More
Set Theory
Aims and objectives
In this lesson, we have discussed about set theory and Venn-Euler Diagrams.
After reading this lesson, you should able to
- About set and methods of description.
- Know about the types of sets.
- Represent sets using Venn-Euler Diagrams and to solve problems.
Introduction
The concept of set theory, originated in 1895 by the German mathematician G.Cantor, is used in various disciplines. This chapter introduces the notation and terminology of set theory…….Read More
The inclusion-Exclusion Principle
Aims and objectives
In this lesson, we have discussed about the Inclusion- Exclusion Principle.
After reading this lesson, you should be able to
- Know what do you mean by an Inclusion-Exclusion Principle.
- Apply Inclusion-Exclusion Principle to solve problems.
Introduction
The Inclusion-Exclusion Principle method is used in finding the number of elements in a finite set (also called cardinal number). While applying this principle some terms are Included and some terms are excluded from an expression, so……Read More
Mathematical Logic
Aims and objectives
In this lesson, we have discussed the basis of propositional logic. With this logic, students can found the truth and falsity of the statements.
After reading this lesson, you should able to understand
- What is a propositional logic? and its usage in reasoning.
- Types of statements
- Connectives for forming compound statements
Introduction
Logic, logical thinking, and correct reasoning have wide applications in many fields, including law, psychology, rhetoric, science, and mathematics. While an interesting study can be made of logic in human lives, we shall restrict our attention mainly to logic as it is used in mathematics. This logic………Read More
Tautology, Contradiction and Arguments
Aims and Objectives
The main aim and objective of this lesson is to know about the Tautology, Contradiction and Arguments.
After reading this unit, students should be able to
- Identify the statements that are Tautology or Contradiction.
- Test the validity of this argument
Introduction
In this lesson we are going to discuss about the certain compound propositions that are always true or false irrespective of the truth values of its component propositions. The statement that is always…..Read More
Method of proof and Predicate Calculus
Aims and objectives
In this lesson, we have discussed method of proofs and predicate calculus. With
the proof methods, one can arrive at a conclusion from the given sequence of premises.
After reading this lesson, you should able to understand
- Direct and Indirect proof methods
- Predicate calculus for reasoning.
- Quantifiers and its types.
Introduction
An argument that establishes the truth of a theorem is called a proof. Logic is a tool for the analysis of proofs. In this lesson, we describe some general methods of proof. In the previous lesson we have discussed the propositional logic which is not sufficient for all our logic needs. We need to go beyond the propositional calculus….Read More
Relations
Aims and Objectives
In this lesson, we have discussed about the Product of sets, Binary Relations and Types of relations.
After completing this lesson, you should able to
- Find product of sets.
- Know about the Binary Relations.
- Know the different types of relations.
Introduction
In mathematics and computer science, there are many relations used to establish a relation between pair of objects taken in a definite order. For example, “less than”, “is parallel to”, “is a subset of”, and so on. A relation between two sets can be defined by listing their elements as ordered pairs. In this lesson…..Read More
Functions
12.0 Aims and Objectives
In this lesson, we have discussed about the Properties of relations, Partial order relations, Equivalence relation, and Composition of relations.
After completing this lesson, you should able to
- Know about Properties of relations.
- Work with Partial order relations.
- Compute Equivalence relation and Composition of relations .
12.1Introduction
Many everyday phenomena involve two quantities that are related to each other by some rule of correspondence. Such a rule of correspondence is called a function. The concept of a function is extremely important in discrete mathematics. Functions are used in the definition of such discrete structures as sequences and strings. Functions are…..Read More
Graph Theory
Aims and objectives
There are various types of graphs, each with its own definition. Unfortunately, some people apply the term “graph” rather loosely, so you can’t be sure what type of graph they’re talking about unless you ask them. After you have finished this chapter, we expect you to use the terminology carefully, not loosely.
Introduction
In mathematics and computer science….Read More
Trees
15.0 Aims and objectives
The main aim and objective of this lesson is know about the trees, its properties,
binary trees, tree traversing and representation of trees in computer.
After reading this unit, you should be able to
- Construct a tree.
- Perform inorder, preorder and postorder tree traversing
- Know about the ways of representing the trees in a…..Read More
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