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Algebra

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  • Difficulty Level :Medium
  • Last Updated :30 May, 2022
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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.

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. Polynomials in One Variable
  2. Zeroes of a polynomial
  3. Remainder Theorem
  4. Multiplying Polynomials
  5. Algebraic Identities of Polynomials
  6. Geometrical meaning of the Zeroes of polynomial
  7. Relationship between Zeroes and coefficients of a polynomial
  8. Division Algorithm for polynomials
  9. Division Algorithm Problems and Solutions

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. Types of Polynomials
  6. Multiplying Polynomials
  7. Standard Algebraic Identities
  8. Factorization
  9. Introduction to factorization
  10. Division of Algebraic Expressions
  11. Linear Equations in One Variable
  12. Solve Linear Equations with Variable on both Sides
  13. Solving Equations which have Linear Expressions on one Side and Numbers on the other Side
  14. Reducing Equations to Simpler Form
  15. Linear Equations and their solutions
  16. Graph of Linear Equations in Two Variables
  17. Equations of Lines Parallel to the x-axis and y-axis
  18. Pair of Linear Equations in Two Variables
  19. Number of Solutions to a System of Equations Algebraically
  20. Graphical methods of Solving a Pair of Linear Equations
  21. Algebraic methods of Solving a Pair of Linear Equations
  22. Equation Reducible to a Pair of linear equations in two variables
  23. Quadratic Equations
  24. Solution of a Quadratic equation by different methods
  25. Roots of a Quadratic Equation
  26. Complex Numbers
  27. Algebra of Real Functions
  28. Algebraic Operations on Complex Numbers
  29. Argand plane and polar representation
  30. Absolute Value of a Complex Number
  31. Imaginary Numbers
  32. Complex Conjugate
  33. Compound Inequalities
  34. Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation
  35. Graphical Solution of Linear Inequalities in Two variables
  36. Solving Linear Inequalities Word Problems
  37. Fundamental Principle of Counting
  38. Introduction to Arithmetic Progressions
  39. Common difference and Nth term
  40. Sum of First n Terms
  41. Binomial Theorem for positive integral indices
  42. Pascal’s Triangle
  43. Introduction to Sequences and Series
  44. General and Middle Terms – Binomial Theorem
  45. Arithmetic Series
  46. Arithmetic Sequences
  47. Geometric Sequence and Series
  48. Geometric Series
  49. Arithmetic Progression and Geometric Progression
  50. 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. Vector Algebra
  2. Dot and Cross Product of two vectors
  3. How to Find the Angle Between Two Vectors?
  4. 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 and functions
  2. Types of Functions
  3. Composite functions
  4. Invertible Functions
  5. Composition of Functions
  6. Inverse Functions
  7. Verifying Inverse Functions by Composition
  8. Introduction to Domain and Range
  9. Piecewise Function
  10. Range of a Function

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. Elementary Operations on Matrice
  8. Inverse of a Matrix by Elementary Operations
  9. Invertible Matrices
  10. Determinants
  11. Properties of Determinants
  12. Area of a Triangle using Determinants
  13. Minors and Cofactors
  14. Adjoint of a Matrix
  15. 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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