ALGEBRA

Elementary Algebra covers the traditional topics studied in a modern elementary algebra course. Arithmetic includes numbers along with mathematical operations like +,  -,  x,  ÷. But in algebra, the numbers are often represented by the symbols and are called variables such as x, a, n, y. It also allows the common formulation of the laws of arithmetic such as, a + b = b + a and it is the first step that shows the systematic exploration of all the properties of a system of real numbers.

The concepts coming under elementary algebra include variables, evaluating expressions and equations, properties of equalities and inequalities, solving the algebraic equations and linear equations having one or two variables, etc.

This is the intermediate level of Algebra. This algebra has a high level of equations to solve as compared to pre-algebra. Advanced algebra will help you to go through the other parts of algebra such as:

• Equations with inequalities
• Matrices
• Solving system of linear equations
• Graphing of functions and linear equations
• Conic sections
• Polynomial Equation
• Quadratic Functions with inequalities
• Polynomials and expressions with radicals
• Sequences and series
• Rational expressions
• Trigonometry
• Discrete mathematics and probability

Abstract algebra is one of the divisions in algebra which discovers the truths relating to algebraic systems independent of the specific nature of some operations. These operations, in specific cases, have certain properties. Thus we can conclude some consequences of such properties. Hence this branch of mathematics called abstract algebra.

Abstract algebra deals with algebraic structures like the fields, groups, modules, rings, lattices, vector spaces, etc.

The concepts of the abstract algebra are below-

1. Sets – Sets is defined as the collection of the objects that are determined by some specific property for a set. For example – A set of all the 2×2 matrices, the set of two-dimensional vectors present in the plane and different forms of finite groups.
2. Binary Operations – When the concept of addition is conceptualized, it gives the binary operations. The concept of all the binary operations will be meaningless without a set.
3. Identity Element – The numbers 0 and 1 are conceptualized to give the idea of an identity element for a specific operation. Here, 0 is called the identity element for the addition operation, whereas 1 is called the identity element for the multiplication operation.
4. Inverse Elements – The idea of Inverse elements comes up with a negative number. For addition, we write “-a” as the inverse of “a” and for the multiplication, the inverse form is written as “a-1″.
5. Associativity – When integers are added, there is a property known as associativity in which the grouping up of numbers added does not affect the sum. Consider an example, (3 + 2) + 4 = 3 + (2 + 4)

Linear algebra is a branch of algebra that applies to both applied as well as pure mathematics. It deals with the linear mappings between the vector spaces. It also deals with the study of planes and lines. It is the study of linear sets of equations with transformation properties. It is almost used in all areas of Mathematics. It concerns the linear equations for the linear functions with their representation in vector spaces and matrices. The important topics covered in linear algebra are as follows:

• Linear equations
• Vector Spaces
• Relations
• Matrices and matrix decomposition
• Relations and Computations

 Commutative algebra is one of the branches of algebra that studies the commutative rings and their ideals. The algebraic number theory, as well as the algebraic geometry, depends on commutative algebra. It includes rings of algebraic integers, polynomial rings, and so on. Many other mathematics areas draw upon commutative algebra in different ways, such as differential topology, invariant theory, order theory, and general topology. It has occupied a remarkable role in modern pure mathematics