Engineering Mathematics

Engineering Mathematics13 Aug 2026 · 22 min read

Properties of Matrix Representations

The kernel of a linear transformation is isomorphic to the null space of its matrix representation, and the range to the column space; rank and nullity transfer.

Engineering Mathematics13 Aug 2026 · 21 min read

Properties of Similar Matrices

Similarity is an equivalence relation preserving the characteristic polynomial, eigenvalues, trace, determinant and rank; equal spectra do not imply similarity.

Properties of Vector Spaces

Theorems derived from the vector space axioms alone: uniqueness of the zero vector and inverses, the zero-scalar rules and the three cancellation laws.

Published Tables of Fields and Curves

Standard tables of number fields, elliptic curves, class numbers and factorisations, and how to use them responsibly.

Quadratic Field Discriminants and Integral Bases

Discriminants and integral bases of quadratic fields, given by closed formulas with no computation required.

Quadratic Residues

Quadratic residues modulo a prime, the exact split into residues and non-residues, and the group-theoretic reason for it.

Quadratic Sieve Factor Base Selection

Choosing the factor base for the quadratic sieve, the quadratic residue criterion, and the multiplier.

Quotient Groups

Constructing the quotient group from a subgroup, well-definedness of the induced operation, and the standard examples.

Radical Computation and the Ring of Multipliers

Computing the p-radical of an order as a kernel, and the ring of multipliers that enlarges the order.

Random Variables

Random variables, their distributions, joint behaviour and independence.

Engineering Mathematics13 Aug 2026 · 22 min read

Range of a Linear Transformation

The range of a linear transformation: definition, why it is a subspace of the codomain, its identity with the column space, and the surjectivity criterion.

Rank and Nullity of a Linear Transformation

Rank and nullity of a linear transformation: dimensions of the range and kernel, the rank-nullity theorem, and dimension tests for injectivity and surjectivity.

Rank and Nullity of a Matrix

Rank is the dimension of the column space, nullity the dimension of the null space; both come from reduced row-echelon form and sum to the column count.

Rank and Nullity of a Nonsingular Matrix

A square matrix is nonsingular exactly when its rank equals its size and its nullity is zero, with the consolidated list of nonsingular matrix equivalences.

Engineering Mathematics13 Aug 2026 · 24 min read

Ranks and Transposes

A matrix and its transpose always have the same rank, so row rank equals column rank, while their nullities generally differ by the difference in shape.

Rational Function Reconstruction

Recovering a rational function from a residue modulo a polynomial, with degree bounds replacing size bounds.

Rational Function Reconstruction in Symbolic Algebra

Using rational function reconstruction inside computer algebra for exact computation over function fields.

Rational Reconstruction

Recovering a rational number from its residue modulo n, the uniqueness conditions, and the role of the extended Euclidean algorithm.

Rational Reconstruction in Symbolic Algebra

Applying rational reconstruction inside computer algebra systems for exact linear solving, interpolation and gcd computation.

Recovering Abelian Group Structure from a Relation Matrix

Recovering the structure and explicit generators of a finite abelian group from a matrix of relations among a generating set.

Engineering Mathematics13 Aug 2026 · 21 min read

Recycling Definitions in Abstract Vector Spaces

How linear combination, span, relations of linear dependence and linear independence transfer unchanged from column vectors to any abstract vector space.

Engineering Mathematics13 Aug 2026 · 19 min read

Reduced Row-Echelon Form

Definition, uniqueness and engineering use of reduced row-echelon form (RREF): leading ones, pivot columns, zero rows and the canonical form of a matrix under row operations.

Reducing the Error Probability

Amplifying the success probability of a randomised algorithm by independent repetition, for one-sided and two-sided error.

Reduction of Indefinite Forms and the Cycle Structure

Reduction of indefinite binary forms, the cycle of reduced forms in each class, and how the cycle encodes the regulator.

Reduction of Positive Definite Binary Forms

Reducing a positive definite binary quadratic form to the unique reduced form in its class, and the resulting class number algorithm.

Regulator and Fundamental Unit Recovery

Extracting fundamental units and the regulator from the kernel of the relation matrix, and confirming the unit system is fundamental.

Relation Matrix Construction

Generating relations among ideal classes, assembling the sparse matrix, and knowing when enough relations have been collected.

Representing Large Integers

Positional representation of multiprecision integers, base selection, sign handling and normalisation invariants.

Engineering Mathematics13 Aug 2026 · 9 min read

Residue Classes and the Ring of Integers Modulo n

The ring Z_n of residue classes, its units and zero divisors, and the condition under which it is a field.

Resultants and Discriminants

The resultant as a criterion for common roots, the discriminant as a test for repeated roots, and how both are computed in practice.

Reversed Formal Laurent Series

Reversed formal Laurent series, the valuation by degree, and their role in rational function reconstruction.

Ring Homomorphisms and Isomorphisms

Ring homomorphisms, kernels as ideals, and the first isomorphism theorem for rings.

Rings: Definitions, Properties and Examples

Commutative rings with unity: axioms, units, and the standard examples used throughout the subject.

Root Finding over Finite Fields

Finding roots of a polynomial in a finite field by GCD with the Frobenius polynomial followed by probabilistic splitting.

Root Finding over the Reals and Complex Numbers

Numerical root finding for polynomials with exact coefficients, root isolation over the reals, and the precision required to be reliable.

Engineering Mathematics13 Aug 2026 · 24 min read

Row Space of a Matrix

The row space of a matrix: definition via the transpose, invariance under row operations, a basis from the non-zero rows of the reduced row-echelon form, and span simplification.

Schoof's Point Counting Algorithm

Schoof's polynomial-time algorithm for counting points on a curve over a finite field, and the SEA improvements.

Schoolbook and Karatsuba Multiplication

Quadratic schoolbook multiplication, the Karatsuba three-multiplication identity, and where the crossover between them sits.

Shanks's Class Group Factoring Method

Shanks's method factoring an integer by finding an ambiguous form in the class group of the corresponding discriminant.

Shanks's Square Forms Factorisation (SQUFOF)

SQUFOF: factoring by finding a square form in the cycle of an indefinite quadratic form, and why it excels for small inputs.

Engineering Mathematics13 Aug 2026 · 22 min read

Similar Matrices

Similar matrices satisfy A = S inverse B S for a non-singular S: definition, a worked similarity transformation, change of basis and the invariants preserved.

Smooth Numbers

Smooth numbers, their density, and why they are the raw material of subexponential factoring and index calculus.

Smoothness and Sub-exponential Complexity

Smooth numbers, the Dickman function, and how balancing smoothness probability against factor base size produces sub-exponential running times.

Solving Linear Congruences

Solving ax = b (mod n): the solvability criterion, the exact number of solutions, and the algorithm via extended Euclid.

Solving Quadratic Congruences

Reducing a general quadratic congruence to a square root extraction, and handling the degenerate cases the reduction assumes away.

Solving Sparse Linear Systems

Iterative methods for large sparse systems over finite fields, and their role as the bottleneck of sieve algorithms.

Solving Systems of Linear Equations

Solving linear systems over a field: consistency, the structure of the solution set, and modular methods for exact rational answers.

Sophie Germain Primes

Sophie Germain primes and safe primes, their use in discrete logarithm cryptography, and the conjectural nature of their density.