In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this...
The preimage of the zero vector under a linear function f is called kernel or null space. In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... The null space (also nullspace) of a matrix A is the set of all vectors v which solve the equation Av = 0, a linear subspace of the space of all vectors. ...
Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, the sciences, and engineering.
Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors which are quantities with a magnitude and a direction, usually depicted as arrows.
Given a vector space V, a nonempty subset W of V which is closed under addition and scalar multiplication is called a subspace of V. Subspaces of V are vector spaces (over the same field) in their own right.
A semi normed vector space is a pair (V,p) where V is a vector space and p a semi norm on V.
A surjective isometry between the normed vector spaces V and W is called a isometric isomorphism, and V and W are called isometrically isomorphic.
The definition of many normed spaces (in particular, Banach spaces) involves a seminorm defined on a vector space and then the normed space is defined as the quotient space by the subspace of elements of seminorm zero.
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