Linear independenceIn linear algebra, a set of vectors is said to be linearly independent if there exists no vector in the set that is equal to a linear combination of the other vectors in the set. If such a vector exists, then the vectors are said to be linearly dependent. Linear independence is part of the definition of linear basis. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors.
Geometric latticeIn the mathematics of matroids and lattices, a geometric lattice is a finiteatomisticsemimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumption of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite, or finite and infinite, matroids, and every geometric or matroid lattice comes from a matroid in this way.
Vector spaceAlgebraic structure in linear algebraVector addition and scalar multiplication: a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.In mathematics, a vector space (also called a linear space) is a set whose elements, often called vectors, can be added together and multiplied ("scaled") by numbers called scalars.
Graph theoryArea of discrete mathematicsA graph with 6 vertices and 7 edgesIn mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called arcs, links, or lines).
Combinatorial optimizationCombinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem.
Partially ordered setMathematical set with an orderingY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Linear algebraLinear algebra is the branch of mathematics concerning linear equations such as a 1 x 1 + ⋯ + a n x n = b , {\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,} linear maps such as the following: ( x 1 , … , x n ) ↦ a 1 x 1 + ⋯ + a n x n , {\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},} and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics.
Network theoryIn mathematics, computer science, and network science, network theory is a part of graph theory. It defines networks as graphs where the vertices or edges possess attributes. Network theory analyses these networks over the symmetric relations or asymmetric relations between their (discrete) components.
TopologyTopology (from the Greek words , 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuousdeformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.CombinatoricsCombinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles.
GeometryBranch of mathematics GeometryProjecting a sphere to a plane Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry ConceptsFeaturesDimension Straightedge and compass constructions Angle Polytope Centroid Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry Zero-dimensional Point One-dimensional Line Line segment Ray Curve Geodesic Length Two-dimensional Surface Plane Area Polygon Simple Convex Concave Star Regular Reuleaux Triangle Centers Altitude Hypotenuse Pythagorean theorem Circular Reuleaux Hyperbolic Ideal Spherical Quadrilateral Parallelogram Rectangle Square Rhombus Rhomboid Trapezoid Kite Circle Radius Diameter Circumference Disk Area Three-dimensional Surface area Volume Polyhedron Platonic Solid Tetrahedron Reuleaux cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Star Toroidal Ideal Solid of revolution Sphere Great circle Cylinder Cone Four - other-dimensional 4-polytope Simplex 5-cell Hypercube Tesseract n-sphere Geometers by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Chern Present day Atiyah Gromov vte Geometry is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.
Axiomatic systemIn mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science. It consists of a set of formal statements known as axioms that are used for the logical deduction of other statements. In mathematics these logical consequences of the axioms may be known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems.*As an Amazon Associate I earn from qualifying purchases.