Dimension (vector space) in the context of Cardinality

In mathematics, 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 operations of vector addition and scalar multiplication. A standard example of a linear map is an ${\displaystyle m\times n}$ matrix, which takes vectors in ${\displaystyle n}$-dimensions into vectors in ${\displaystyle m}$-dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by scalars.

A linear map is a homomorphism of vector spaces. Thus, a linear map ${\displaystyle T:V\to W}$ satisfies ${\displaystyle T(ax+by)=aTx+bTy}$, where ${\displaystyle a}$ and ${\displaystyle b}$ are scalars, and ${\displaystyle x}$ and ${\displaystyle y}$ are vectors (elements of the vector space ${\displaystyle V}$). A linear mapping always maps the origin of ${\displaystyle V}$ to the origin of ${\displaystyle W}$, and linear subspaces of ${\displaystyle V}$ onto linear subspaces in ${\displaystyle W}$ (possibly of a lower dimension); for example, it maps a plane through the origin in ${\displaystyle V}$ to either a plane through the origin in ${\displaystyle W}$, a line through the origin in ${\displaystyle W}$, or just the origin in ${\displaystyle W}$. Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors. The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.

In geometry, there exist various rotation formalisms to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational (or angular) kinematics is the science of quantitative description of a purely rotational motion. The orientation of an object at a given instant is described with the same tools, as it is defined as an imaginary rotation from a reference placement in space, rather than an actually observed rotation from a previous placement in space.

According to Euler's rotation theorem, the rotation of a rigid body (or three-dimensional coordinate system with a fixed origin) is described by a single rotation about some axis. Such a rotation may be uniquely described by a minimum of three real parameters. However, for various reasons, there are several ways to represent it. Many of these representations use more than the necessary minimum of three parameters, although each of them still has only three degrees of freedom.

In mathematics, an inner product space is a real or complex vector space endowed with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in ${\displaystyle \langle a,b\rangle }$. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates. Inner product spaces of infinite dimensions are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.

An inner product naturally induces an associated norm, (denoted ${\displaystyle |x|}$ and ${\displaystyle |y|}$ in the picture); so, every inner product space is a normed vector space. If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space ${\displaystyle {\overline {H}}.}$ This means that ${\displaystyle H}$ is a linear subspace of ${\displaystyle {\overline {H}},}$ the inner product of ${\displaystyle H}$ is the restriction of that of ${\displaystyle {\overline {H}},}$ and ${\displaystyle H}$ is dense in ${\displaystyle {\overline {H}}}$ for the topology defined by the norm.

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group ${\displaystyle \operatorname {Aut} (X)}$ is the group consisting of all group automorphisms of X.

Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group.

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ${\displaystyle V}$ with finite dimension is a basis for ${\displaystyle V}$ whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for ${\displaystyle \mathbb {R} ^{n}}$ arises in this fashion.An orthonormal basis can be derived from an orthogonal basis via normalization.The choice of an origin and an orthonormal basis forms a coordinate frame known as an orthonormal frame.

For a general inner product space ${\displaystyle V,}$ an orthonormal basis can be used to define normalized orthogonal coordinates on ${\displaystyle V.}$ Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces the study of a finite-dimensional inner product space to the study of ${\displaystyle \mathbb {R} ^{n}}$ under the dot product. Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram–Schmidt process.

In mathematics, specifically linear algebra, a degenerate bilinear form ${\displaystyle f(x,y)}$ on a vector space ${\displaystyle V}$ is a bilinear form such that the map from ${\displaystyle V}$ to ${\displaystyle V^{*}}$ (the dual space of ${\displaystyle V}$) given by ${\displaystyle v\mapsto (x\mapsto f(x,v))}$ has a non-trivial kernel, i.e. there exist some non-zero ${\displaystyle x}$ in ${\displaystyle V}$ such that ${\displaystyle f(x,y)=0}$ for all ${\displaystyle y\in V}$.

An equivalent definition when ${\displaystyle V}$ is finite-dimensional is that the previous map is not an isomorphism.