Algebra

Matrices, Theory of Equation and Group Theory

Authors: Dr. Vimal Saraswat, Dr. Gajendrapal Singh Rathore

This book includes the following topics

Matrix and its Rank

• Introduction
• Types of matrices
• Operation on matrices or matrix algebra
• Properties of matrix addition
• Properties of scalar multiplication of matrix
• Trace of a matrix
• Multiplication of matrices
• Properties of matrix multiplication
• Transpose matrix
• Symmetric and skew symmetric matrix
• Orthogonal matrix
• Conjugate matrix
• Conjugate transpose matrix or tranjugate matrix
• Unitary matrix
• Hermitian and skew-Hermitian matrices
• Sub matrix of a matrix
• Determinant of a square matrix
• Properties of determinant
• Singular and non-singular matrix
• Adjoint or adjugate matrix
• Inverse of a matrix
• Adjoint method for finding the inverse of a matrix
• Elementary transformation method for finding the inverse of a matrix
• Vector
• Linear combination
• Linear independent and linear dependent vector
• Rank of a matrix
• Column rank and row rank of a matrix
• Nullity of a matrix
• Equivalent matrix
• Normal or canonical form of a matrix

Linear Equations, Eigen Values and Eigen Vectors

• Introduction
• System of homogeneous linear equation and their solution
• System of non-homogeneous linear equations and their solution

Cramer’s rule, Inverse matrix method, Elementary transformation method

• Condition of consistency
• Working method for finding the solution of system of non-homogeneous linear equations
• Triangular or echelon form of a matrix
• Characteristics equation of a matrix
• Cayley-Hamilton theorem
• Application of Cayley-Hamilton’s theorem
• Eigen values and eigen vectors
• Properties of eigen values of any characteristic equation
• Nature of characteristic root or eigen values of some special type of matrices

Theory of Equations

• Introduction
• Rational integral function or polynomials
• Equation
• Roots of an equation
• Identity
• Properties of equation
• Relation between roots and coefficients of equation
• Solution of equation using relations of roots and coefficients
• Symmetric function of the roots
• Transformation of equations

Equation whose roots are k times the roots of a given equation, Equation whose roots are oppositely signed of a given equation, Equation whose roots are the reciprocals of the roots of a given equation

• Reciprocal equation
• Synthesis division

To diminish the roots by a given number; To remove a particular terms of an equation

Cubic and Biquadratic Equations

• Introduction
• Descarte’s rule of sign
• Solution of cubic equation by Cardon’s method
• Solution of general cubic equation by Cardon’s methods
• Nature of the roots of cubic equations
• Solution of biquadratic equation by Ferrari’s method
• Newton’s method of approximation
• Horner’s method

Group

• Introduction
• Cartesian product
• Algebraic structure
• Binary operation
• Group
• Finite and infinite group
• Groupoid or quasi group
• Semi-group
• Monoid
• Commutative or abelian group
• Order of a group
• Modulo operation
• Uniqueness property in group
• Properties of inverse in group
• Cancellation law in group
• Left and right identity element
• Left and right inverse
• Theorems based on semi group
• Integral power of an element of group
• Order of an element of a group
• Klein’s 4-group
• Theorems based on order of element of a group

Subgroup

• Introduction
• Subgroup
• Examples of subgroup
• Multiplication of complexes of a group
• Inverse of a complex of a group
• Necessary and sufficient condition for a subgroup
• Necessary and sufficient condition for a finite subset to be a subgroup
• Necessary and sufficient condition for union to be a subgroup
• Coset
• Lagrange’s theorem
• Index of subgroup
• Relation of congruence modulo of a group with respect to a subgroup
• Euler’s ф function
• Properties of Euler’s ф function
• Euler’s theorem
• Fermat’s theorem

Cyclic Group

• Introduction
• Cyclic group
• Theorems based on cyclic group

Normal Subgroup and Quotient Group

• Introduction
• Normal subgroup
• Simple group
• Theorem based on normal subgroup
• Quotient group
• Commutator subgroup

Permutation Group

• Introduction
• Permutation
• Equivalent or equal permutation
• Identity permutation)
• Product of permutation
• Inverse permutation
• Permutation group
• Symmetric group
• Cyclic permutation or cycles and the length of cyclic permutation
• Inverse of cycle
• Order of a cycle
• Disjoint cycles
• Order of permutation
• Transposition
• Even and odd permutation

Group Homomorphism

• Introduction
• Homomorphism
• Types of homomorphism

Monomorphism, Epimorphism, Isomorphism, Endomorphism, Automorphism

• Kernel of homomorphism
• Theorem based on homomorphism
• Natural homomorphism
• Fundamental theorem of homomorphism
• Three theorems on isomorphism
• Cayley’s theorem
• Inner automorphism

Centre of group, Normalizer and Class equation

• Introduction
• Conjugate elements
• Conjugate class
• Self-conjugate element
• Normalizer of an element
• Centre of group