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Algebra

  • Last Updated : 04 Jun, 2021

Algebra is a branch of mathematics that deals with number theory, geometry, and analysis. It is one of the oldest branches in the history of mathematics. Algebra is defined as the analysis of mathematical symbols and laws, as well as the manipulation of these mathematical symbols, according to some definitions. Algebra covers a wide range of topics, from solving basic problems to studying abstractions. Algebra equations are included in several chapters of mathematics that students will study in school. In addition, algebra contains a number of formulas and identities.

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The definition of algebra is focused on variables, which are undefined values. Equations are a crucial concept in algebra. To execute arithmetic operations, it follows a set of laws. The laws are used to interpret data sets that have two or more variables. It is used to analyze a variety of topics around us. Basic algebra, abstract algebra, linear algebra, advanced algebra, and commutative algebra are some of the sub-branches of algebra are discussed below:

Table of Contents

  1. Polynomials
  2. Linear Equations
  3. Set theory
  4. Vectors
  5. Relations and Functions
  6. Matrices and Determinants
  7. Permutation and Combination
  8. Mathematical Induction

Linear Algebra

Linear algebra is a type of algebra that can be used in both applied and pure mathematics. It is concerned with linear mappings between vector spaces. It also entails the investigation of planes and tracks. It involves the investigation of linear sets of equations with transformation properties. It is concerned with linear equations and their representation in vector spaces and by matrices for linear functions. The following are some of the most important parts discussed in linear algebra:

Polynomials 



Polynomials are made by variables and coefficients, which are algebraic expressions. Indeterminate is another name for variables. For polynomial expressions, we may perform arithmetic operations such as addition, subtraction, multiplication, and positive integer exponents, but not division by variable. Following are the topics that discuss polynomials deeply:

  1. Zeroes of a polynomial
  2. Remainder Theorem
  3. Multiplying Polynomials
  4. Algebraic Identities of Polynomials
  5. Geometrical meaning of the Zeroes of polynomial
  6. Relationship between Zeroes and coefficients of a polynomial
  7. Division Algorithm for polynomials

Linear Equations 

Linear equations are first-order equations. In the coordinate system, these equations are defined for lines. A linear equation is an equation for a straight line. The straight-line equation is written as y = mx + b, where m denotes the line’s slope and b denotes the y-intercept. The first-order equations are known as linear equations. In the coordinate system, these equations are defined for lines. Since the highest exponent of variables is 1, linear equations are also known as first-degree equations. Following are the topics that discuss linear equations deeply:

  1. Standard Algebraic Identities
  2. Algebraic expressions
  3. Like and Unlike Algebraic Terms
  4. Mathematical Operations on Algebraic Expressions
  5. Standard Algebraic Identities
  6. Factorization
  7. Introduction to factorization
  8. Division of Algebraic Expressions
  9. Linear Equations in One Variable
  10. Solve Linear Equations with Variable on both Sides
  11. Solving Equations which have Linear Expressions on one Side and Numbers on the other Side
  12. Reducing Equations to Simpler Form
  13. Linear Equations and their solutions
  14. Pair of Linear Equations in Two Variables
  15. Graphical methods of Solving a Pair of Linear Equations
  16. Algebraic methods of Solving a Pair of Linear Equations
  17. Equation Reducible to a Pair of linear equations in two variables
  18. Quadratic Equations
  19. Solution of a Quadratic equation by different methods
  20. Roots of a Quadratic Equation
  21. Algebraic Operations on Complex Numbers
  22. Argand plane and polar representation
  23. Imaginary Numbers
  24. Compound Inequalities
  25. Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
  26. Graphical Solution of Linear Inequalities in Two variables
  27. Fundamental Principle of Counting
  28. Introduction to Arithmetic Progressions
  29. Common difference and Nth term
  30. Sum of First n Terms
  31. Binomial Theorem for positive integral indices
  32. Pascal’s Triangle
  33. Introduction to Sequences and Series
  34. General and Middle Terms – Binomial Theorem
  35. Arithmetic Series
  36. Arithmetic Sequences
  37. Geometric Sequence and Series
  38. Special Series

Abstract Algebra

Abstract algebra is one of the branches of algebra that finds truths about algebraic structures regardless of the existence of certain operations. In certain instances, these operations have special properties. As a result, we can draw certain conclusions about the implications of those properties. As a result, abstract algebra is a subset of mathematics. Groups, sets, modules, rings, lattices, vector spaces, and other algebraic structures are studied in abstract algebra. Following are the list of the articles that discuss the same:

Set Theory

Set theory is a branch of mathematical logic that investigates sets, which are arrays of objects informally. While any object can be gathered into a group, set theory as a branch of mathematics is mostly concerned with those that are important to all mathematics. Sets are an ordered array of items in mathematics that can be expressed in set-builder or roster form. Sets are usually denoted by curly braces;{} for example, A = {1,2,3,4} is a collection. Let’s learn more about the sets in the following articles:

  1. Sets and their representations
  2. Different kinds of Sets
  3. Subsets, Power Sets, and Universal Sets
  4. Venn Diagrams
  5. Operations on Sets
  6. Union and Intersection of sets
  7. Cartesian Product of Sets

Vectors

One of the most important aspects of algebra is vector algebra. It is a course that focuses on the algebra of vector quantities. There are two kinds of measurable quantities: scalars and vectors, as we all know. The magnitude of a scalar quantity is the only thing that matters, while the magnitude and direction of a vector quantity are also essential. A vector is a two-dimensional object of both magnitude and direction. It is normally represented by an arrow with a length (→) that indicates the magnitude and direction. A vector is a two-dimensional object of both magnitude and direction. It is normally represented by an arrow with a length that indicates the magnitude and direction. The arrowhead on the vector arrow has an arrowhead, and the tail on the other end is the tail. It is denoted by the letter V. The vector’s magnitude is denoted by the letter |V|. Let’s discuss more the vector and its algebra in the following articles:



  1. Dot and Cross Product of two vectors
  2. Section Formula

Advanced Algebra

There are advanced styles of algebra taught in schools such as abstract algebra, linear algebra, and standard algebra, in addition to basic algebra. This involves learning how to use a matrix to solve a large number of linear equations at the same time. The analysis of objects contained in equations, moving beyond numbers to the more abstract of classes of numbers, is known as abstract algebra. Let’s discuss more of it in the following topics:

Relations and Functions

The most significant subjects in mathematics are “Relations and Functions.” Relations and functions are two distinct terms that have different mathematical interpretations. One might be puzzled by the differences between them. Before we go even further, let’s look at a clear example of the differences between the two. An ordered pair is represented as (INPUT, OUTPUT): The relation shows the relationship between INPUT and OUTPUT. Whereas, a function is a relation that derives one OUTPUT for each given INPUT. Let’s discuss more of the topic in the following articles:

  1. Relations
  2. Functions
  3. Types of Functions
  4. Composite functions
  5. Invertible Functions
  6. Introduction to Domain and Range
  7. Algebraic Operations on Complex Numbers
  8. Argand plane and polar representation

Matrices and Determinants

In linear algebra, determinants and matrices are used to solve linear equations by applying Cramer’s law to a series of non-homogeneous linear equations. Only square matrices are used to measure determinants. While a matrix’s determinant is empty, it’s known as a singular determinant, and when it’s one, it’s known as unimodular. The determinant of the matrix must be nonsingular, that is, its value must be nonzero, for the set of equations to have a unique solution. Let us look at the definitions of determinants and matrices, as well as the various forms of matrices and their properties, using examples in the following articles:

  1. Matrices and its Types
  2. Mathematical Operations on Matrices
  3. Properties of Matrix Addition and Scalar Multiplication
  4. How to Multiply Matrices
  5. Transpose of a matrix
  6. Symmetric and Skew Symmetric Matrices
  7. Inverse of a Matrix by Elementary Operations
  8. Invertible Matrices
  9. Determinants
  10. Properties of Determinants
  11. Area of a Triangle using Determinants
  12. Minors and Cofactors
  13. Adjoint of a Matrix
  14. Application of Determinants and Matrices

Permutations and Combinations

Permutation and combination are methods for representing a collection of objects by choosing them from a list and dividing them into subsets. It specifies the different methods for organizing a set of data. Permutations are used to choose data or events from a group, while combinations are used to represent the order in which they are represented. Let’s discuss them deeply in the following articles:

  1. Permutation
  2. Combinations

Mathematical Induction

For each and every natural number n, mathematical induction is a technique for proving a proposition, hypothesis, or formula that is assumed to be valid. The ‘Principle of Mathematical Induction is a generalization of this that we can use to prove any mathematical statement. Therefore, let’s learn more about it in the following articles:

  1. Principle of Mathematical Induction
  2. Conditional Statements & Implications
  3. Validating Statements

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