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In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis.Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations.
The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below.
Let V be a vector space of dimension n over a field F and let
be an ordered basis for V. Then for every there is a unique linear combination of the basis vectors that equals v:
The coordinate vector of v relative to B is the sequence of coordinates
This is also called the representation of v with respect of B, or the B representation of v. The α-s are called the coordinates of v. The order of the basis becomes important here, since it determines the order in which the coefficients are listed in the coordinate vector.
Coordinate vectors of finite-dimensional vector spaces can be represented by matrices as column or row vectors. In the above notation, one can write
We can mechanize the above transformation by defining a function , called the standard representation of V with respect to B, that takes every vector to its coordinate representation: . Then is a linear transformation from V to Fn. In fact, it is an isomorphism, and its inverse is simply
Alternatively, we could have defined to be the above function from the beginning, realized that is an isomorphism, and defined to be its inverse.
Let P3 be the space of all the algebraic polynomials of degree at most 3 (i.e. the highest exponent of x can be 3). This space is linear and spanned by the following polynomials:
then the coordinate vector corresponding to the polynomial
According to that representation, the differentiation operator d/dx which we shall mark D will be represented by the following matrix:
Using that method it is easy to explore the properties of the operator, such as: invertibility, Hermitian or anti-Hermitian or neither, spectrum and eigenvalues, and more.
The Pauli matrices, which represent the spin operator when transforming the spin eigenstates into vector coordinates.
Let B and C be two different bases of a vector space V, and let us mark with the matrix which has columns consisting of the C representation of basis vectors b1, b2, …, bn:
This matrix is referred to as the basis transformation matrix from B to C. It can be regarded as an automorphism over . Any vector v represented in B can be transformed to a representation in C as follows:
Under the transformation of basis, notice that the superscript on the transformation matrix, M, and the subscript on the coordinate vector, v, are the same, and seemingly cancel, leaving the remaining subscript. While this may serve as a memory aid, it is important to note that no such cancellation, or similar mathematical operation, is taking place.
The matrix M is an invertible matrix and M−1 is the basis transformation matrix from C to B. In other words,
Suppose V is an infinite-dimensional vector space over a field F. If the dimension is κ, then there is some basis of κ elements for V. After an order is chosen, the basis can be considered an ordered basis. The elements of V are finite linear combinations of elements in the basis, which give rise to unique coordinate representations exactly as described before. The only change is that the indexing set for the coordinates is not finite. Since a given vector v is a finite linear combination of basis elements, the only nonzero entries of the coordinate vector for v will be the nonzero coefficients of the linear combination representing v. Thus the coordinate vector for v is zero except in finitely many entries.
The linear transformations between (possibly) infinite-dimensional vector spaces can be modeled, analogously to the finite-dimensional case, with infinite matrices. The special case of the transformations from V into V is described in the full linear ring article.
In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets".
In mathematics, any vector space has a corresponding dual vector space consisting of all linear forms on , together with the vector space structure of pointwise addition and scalar multiplication by constants.
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of vector spaces; in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix multiplication. Representations of groups are important because they allow many group-theoretic problems to be reduced to problems in linear algebra, which is well understood. They are also important in physics because, for example, they describe how the symmetry group of a physical system affects the solutions of equations describing that system.
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.
In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.
In mathematics, an invariant subspace of a linear mapping T : V → V from some vector space V to itself, is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.
In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a rotation group.
In linear algebra, the Gram matrix of a set of vectors in an inner product space is the Hermitian matrix of inner products, whose entries are given by . If the vectors are real and the columns of matrix , then the Gram matrix is .
In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis.
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras.
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid. For the case of a finite-dimensional graph, the discrete Laplace operator is more commonly called the Laplacian matrix.
In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra. The most important examples are those with entries in a commutative superalgebra or an ordinary field.
In mathematics, quasi-bialgebras are a generalization of bialgebras: they were first defined by the Ukrainian mathematician Vladimir Drinfeld in 1990. A quasi-bialgebra differs from a bialgebra by having coassociativity replaced by an invertible element which controls the non-coassociativity. One of their key properties is that the corresponding category of modules forms a tensor category.
In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation. In physics, this concept is applied to classical mechanics where rotational 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.
In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.
In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
There are many ways to derive the Lorentz transformations utilizing a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory.
This is a glossary for the terminology applied in the mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory. Because of the lack of other options, the glossary also includes some generalizations such as quantum group.