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Ring Theory & Linear Algebra Notes

Topics in our Ring Theory & Linear Algebra Notes PDF

In these “Ring Theory & Linear Algebra Notes PDF”, you will study the basic concepts of ring of polynomials and irreducibility tests for polynomials over ring of integers, used in finite fields with applications in cryptography. This course emphasizes the application of techniques using the adjoint of a linear operator and their properties to least squares approximation and minimal solutions to systems of linear equations.

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

Polynomial Rings and Unique Factorization Domain (UFD): Polynomial rings over commutative rings, Division algorithm and consequences, Principal ideal domains, Factorization of polynomials, Reducibility tests, Irreducibility tests, Eisenstein’s criterion, Unique factorization in Z[x]; Divisibility in integral domains, Irreducibles, Primes, Unique factorization domains, Euclidean domains.

Dual Spaces and Diagonalizable Operators: Dual spaces, Double dual, Dual basis, Transpose of a linear transformation and its matrix in the dual basis, Annihilators; Eigenvalues, Eigenvectors, Eigenspaces and characteristic polynomial of a linear operator; Diagonalizability, Invariant subspaces and Cayley−Hamilton theorem; Minimal polynomial for a linear operator.

Inner Product Spaces: Inner product spaces and norms, Orthonormal basis, Gram−Schmidt orthogonalization process, Orthogonal complements, Bessel’s inequality.

Adjoint Operators and Their Properties: Adjoint of a linear operator, Least squares approximation, Minimal solutions to systems of linear equations, Normal, self-adjoint, unitary and orthogonal operators and their properties.


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