In these “AlgebraNotes PDF”, you will study the basic tools of theory of equations, complex numbers, number theory and matrices to understand their connection with the real-world problems. Perform matrix algebra with applications to computer graphics.

The topics we will cover will be taken from the following list:

Theory of Equations and Complex Numbers: Polynomials, The remainder and factor theorem, Synthetic division, Factored form of a polynomial, Fundamental theorem of algebra, Relations between the roots and the coefficients of polynomial equations, Theorems on imaginary, integral and rational roots; Polar representation of complex numbers, De Moivre’s theorem for integer and rational indices and their applications. The nth roots of unity.

Equivalence Relations and Functions: Equivalence relations, Functions, Composition of functions, Invertibility and inverse of functions, One-to-one correspondence and the cardinality of a set.

Basic Number Theory: Well ordering principle, The division algorithm in ℤ, Divisibility and the Euclidean algorithm, Fundamental theorem of arithmetic, Modular arithmetic and basic properties of congruences; Principle of mathematical induction.

Row Echelon Form of Matrices and Applications: Systems of linear equations, Row reduction and echelon forms, Vector equations, The matrix equation Ax = b, Solution sets of linear systems, The inverse of a matrix; Subspaces, Linear independence, Basis and dimension, The rank of a matrix and applications; Introduction to linear transformations, The matrix of a linear transformation; Applications to computer graphics, Eigenvalues and eigenvectors, The characteristic equation and Cayley−Hamilton theorem.

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