Engineering

Independent and Dependent Variables

Pivot columns make variables dependent, non-pivot columns make them free: how to partition the unknowns, count degrees of freedom and describe an infinite solution set exactly.

Infinite Discrete Probability Distributions

Countably infinite sample spaces, convergence conditions, and the geometric distribution arising from unbounded loops.

Engineering Mathematics13 Aug 2026 · 21 min read

Injective Linear Transformations

Injective (one-to-one) linear transformations: the defining condition, how to prove injectivity by reducing to a homogeneous system, and how to disprove it.

Injective Transformations, Independence and Dimension

How injective linear transformations preserve linear independence, why the images of a basis decide injectivity, and why domain dimension cannot exceed codomain.

Engineering Mathematics13 Aug 2026 · 20 min read

Inner Products of Vectors

The complex inner product of two column vectors: its definition with a conjugate on the second argument, additivity, scalar rules, conjugate symmetry and positivity.

Integer Addition and Subtraction

Multiprecision addition and subtraction: carry and borrow propagation, sign handling, and why both are linear.

Integer Division with Remainder

Multiprecision division: the normalisation step, digit estimation, correction, and why division is harder to implement than multiplication.

Integer Kernel and Image via LLL

Computing a reduced basis of the integer kernel and image of a matrix, and why this is not the same as clearing denominators from a rational kernel.

Integer Multiplication

Schoolbook multiplication of multiprecision integers, the accumulator requirement, and where the quadratic cost comes from.

Integer Square Root and Perfect Power Detection

Newton iteration for integer square roots, exact perfect power detection, and why these cheap tests belong at the front of every factoring routine.

Integral LLL: Avoiding Floating Point

Running LLL entirely in integer arithmetic using scaled Gram-Schmidt quantities, and why this matters for exact downstream computation.

Inverse Image and Supplementation of Subspaces

Solving for preimages of a subspace, extending an independent set to a basis, and the linear algebra primitives these support.

Engineering Mathematics13 Aug 2026 · 25 min read

Invertibility Criteria for Linear Transformations

A linear transformation is invertible exactly when it is injective and surjective. Criteria, kernel and range tests, composition rules and the inverse map.

Invertible Linear Transformations

Invertible linear transformations: the identity map, the two-sided composition condition defining an inverse, uniqueness, and why the inverse is itself linear.

Isogenies and Endomorphism Rings

Isogenies as maps of curves respecting the group law, and the two possible endomorphism rings over a field of characteristic zero.

Kernel and Image of a General Matrix

Computing kernel and image bases for rectangular matrices over a field, and why the integer analogue is a different problem.

Kernel of a Linear Transformation

The kernel of a linear transformation: definition, why it is always a subspace, its role as the preimage of zero, and why a trivial kernel means injectivity.

Kernels, Images and the Isomorphism Theorems

The kernel and image of a homomorphism, and the first isomorphism theorem relating them to a quotient.

L-Functions and the Birch-Swinnerton-Dyer Conjecture

The L-function of an elliptic curve, the Birch-Swinnerton-Dyer conjecture, and what can and cannot be computed about rank.

Language Recognition and Complexity Classes

Randomised complexity classes RP, co-RP, BPP and ZPP, and where primality testing sits among them.

Lattice Definitions and Quadratic Forms

Lattices as discrete subgroups, the Gram matrix, and the correspondence between lattices with a basis and positive definite quadratic forms.

Lattice Determinant and the Hadamard Bound

The lattice determinant as a basis-independent invariant, the Hadamard inequality, and using the orthogonality defect to measure basis quality.

Lattices, Complex Tori and the Weierstrass p-Function

The identification of elliptic curves over the complex numbers with complex tori, and the Weierstrass function that realises it.

Learning Pathways in Computational Number Theory

Suggested routes through the collection for cryptography, computer algebra, coding theory and pure mathematics readers.

Learning Pathways Through Computational Number Theory

Four routes through this 182-page collection, chosen by what you need to build rather than by chapter order.

Legendre, Jacobi and Kronecker Symbol Computation

The three quadratic symbols, their differing meanings, and the reciprocity-based algorithm that computes them without factoring.

Lehmer's Accelerated GCD Computation

Lehmer's method: running many GCD steps on single-precision leading digits before touching the full multiprecision operands.

Engineering Mathematics13 Aug 2026 · 23 min read

Linear Algebra in Practice: The Trail Mix Blending Problem

A full blending and production-planning case study: mass balances as linear equations, unique versus infinite solution sets, feasibility bounds and profit optimisation.

Engineering Mathematics13 Aug 2026 · 25 min read

Linear Algebra Notation Reference

A consolidated reference for linear algebra notation: matrices, entries, vectors, the four subspaces, spans, inner products, transformations and matrix representations.

Engineering Mathematics13 Aug 2026 · 15 min read

Linear Algebra Theorem Map — Applications, Practice and Reference

How the theorems of linear algebra depend on one another: from row operations and RREF through span, bases and dimension to rank, determinants and eigenvalues.

Engineering Mathematics13 Aug 2026 · 21 min read

Linear Algebra Theorem Map — Foundations and Core Equivalences

How the theorems of linear algebra depend on one another: from row operations and RREF through span, bases and dimension to rank, determinants and eigenvalues.

Engineering Mathematics13 Aug 2026 · 21 min read

Linear Combinations of Vectors

Linear combinations of column vectors: definition, the equivalence between solving a linear system and writing a constant vector as a combination of columns.

Linear Dependence and Spans

Why a linearly dependent set always contains a vector expressible from the others, and how removing it shrinks a spanning set without changing the span it generates.

Linear Independence and Bases

Linear independence, spanning sets and bases, and the conditions under which a basis exists.

Engineering Mathematics13 Aug 2026 · 24 min read

Linear Independence and Nonsingular Matrices

Why a square matrix is nonsingular exactly when its columns are linearly independent, the chained proof, and the growing list of nonsingular equivalences.

Engineering Mathematics13 Aug 2026 · 20 min read

Linear Independence in Abstract Vector Spaces

Relations of linear dependence and linear independence in any vector space: the definitions, the reduction to a homogeneous system, and worked tests in matrix and polynomial spa…

Engineering Mathematics13 Aug 2026 · 24 min read

Linear Independence of Vectors

Relations of linear dependence, the definition of a linearly independent set of column vectors, and the homogeneous-system and rank tests that decide it.

Engineering Mathematics13 Aug 2026 · 24 min read

Linear Transformations

The definition of a linear transformation between vector spaces: additivity, homogeneity, the zero-to-zero theorem, and how to tell a linear map from an affine one.

Engineering Mathematics13 Aug 2026 · 23 min read

Linear Transformations and Linear Combinations

A linear transformation passes through any linear combination, so its values on a basis determine it everywhere. The superposition theorem and its use in prediction.

Linearly Generated Sequences

Sequences satisfying linear recurrences, their minimal polynomials, and the equivalence with rational generating functions.

LLL for Linearly Dependent Generating Sets

Extending LLL to generating sets that are not independent, and using the resulting zero vectors to extract relations.

LLL Reduction Quality and Proof Sketch

Why LLL terminates in polynomial time and what its output guarantees, via the potential function and the Lovasz condition.

LLL with Deep Insertions

Allowing a vector to move further than one position, the quality gain, and the loss of the polynomial time guarantee.

LLL-Based Hermite Normal Form Computation

Using lattice reduction to control entry growth during Hermite normal form computation, and when this beats the modular approach.

Lucas Sequences and Lucas Pseudoprimes

Lucas sequences, the Lucas probable prime test, and why it complements the strong pseudoprime test rather than duplicating it.

Machine Models and Complexity Theory

The random access machine and Turing machine models, what counts as a primitive operation, and how the choice of model affects stated complexities.

Markov's and Chebyshev's Inequalities

Markov's and Chebyshev's inequalities, their proofs, and how they bound deviation from the mean.

Mathematical Notation and Standing Conventions

The notational conventions, symbol set and standing assumptions used throughout the KEVOS computational number theory collection.