Exterior algebraIn mathematics, the exterior algebra or Grassmann algebra of a vector spaceV{\displaystyle V} is an associative algebra that contains V,{\displaystyle V,} which has a product, called exterior product or wedge product and denoted with ∧{\displaystyle \wedge }, such that v∧v=0{\displaystyle v\wedge v=0} for every vector v{\displaystyle v} in V.{\displaystyle V.} The exterior algebra is named after Hermann Grassmann, and the names of the product come from the "wedge" symbol ∧{\displaystyle \wedge } and the fact that the product of two elements of V{\displaystyle V} is "outside" V.{\displaystyle V.}The wedge product of k{\displaystyle k} vectors v1∧v2∧⋯∧vk{\displaystyle v_{1}\wedge v_{2}\wedge \dots \wedge v_{k}} is called a simple k{\displaystyle k}-vector or k{\displaystyle k}-blade.
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.
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.
Ernst KummerErnst Eduard Kummer (German:; 29 January 1810 – 14 May 1893) was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker. LifeKummer was born in Sorau, Brandenburg (then part of Prussia). His father, a physician, died in 1813.
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.
DimensionProperty of a mathematical space From left to right: a square, a cube and a tesseract. The square is two-dimensional (2D) and bounded by one-dimensional line segments; the cube is three-dimensional (3D) and bounded by two-dimensional squares; the tesseract is four-dimensional (4D) and bounded by three-dimensional cubes. The first four spatial dimensions, represented in a two-dimensional picture.
GrassmannianMathematical spaceIn mathematics, a GrassmannianGrk(V){\displaystyle \mathbf {Gr} _{k}(V)}, also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k{\displaystyle k}-dimensionallinear subspaces of an n{\displaystyle n}-dimensional vector spaceV{\displaystyle V} over a fieldK{\displaystyle K} that has a differentiable structure.PhysicsPhysics is the scientific study of matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. It is one of the most fundamental scientific disciplines. A scientist who specializes in the field of physics is called a physicist. Physics is one of the oldest academic disciplines.
MathematicianA mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. HistoryOne of the earliest known mathematicians was Thales of Miletus (c. 624 – c. 546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed.LinguisticsScientific study of languageLinguistics is the scientific study of language.
PhysicistA physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms.
PolymathA polymath or polyhistor is an individual whose knowledge spans many different subjects, known to draw on complex bodies of knowledge to solve specific problems. Polymaths often prefer a specific context in which to explain their knowledge, but some are gifted at explaining abstractly and creatively.*As an Amazon Associate I earn from qualifying purchases.