Explore: Linear Spaces

requirements, called vector axioms. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers

vector space. All linear maps between finite-dimensional vector spaces are also continuous. An isometry between two normed vector spaces is a linear map

and more specifically in linear algebra, a linear map (or linear mapping) is a particular kind of function between vector spaces, which respects the basic

vector spaces, the concept has been introduced in homological algebra, and it is widely used for graded algebras, which are graded vector spaces with additional

the parent space which retains the same structure. While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological

Hilbert spaces, other generalizations of Euclidean spaces were known to mathematicians and physicists. In particular, the idea of an abstract linear space (vector

representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental

E by the kernel of the associated linear map. This is the first isomorphism theorem for affine spaces. Affine spaces are usually studied by analytic geometry

Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. A Banach space is a

with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces. Basis vectors find applications