We have chosen to give it at the end of the section since it deals with differential equations and thus is not purely linear algebra.

This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

* Given any Banach space X, the continuous linear operators A: X → X form a unitary associative algebra ( using composition of operators as multiplication ); this is a Banach algebra.

That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor – Dedekind axiom.

* Basis ( linear algebra )

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f < sup > x </ sup > and f < sub > y </ sub > are all linear transformations.

Is X a Banach space, the space B ( X ) = B ( X, X ) forms a unital Banach algebra ; the multiplication operation is given by the composition of linear maps.

The mathematical structure of quantum mechanics is based in large part on linear algebra:

* The algebra of all continuous linear operators on a Banach space E ( with functional composition as multiplication and the operator norm as norm ) is a unital Banach algebra.

When the Banach algebra A is the algebra L ( X ) of bounded linear operators on a complex Banach space X ( e. g., the algebra of square matrices ), the notion of the spectrum in A coincides with the usual one in the operator theory.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

The linear transformation is also called the curvature transformation or endomorphism.

For an abstract vector space V ( rather than the concrete vector space ), or more generally a module M over a ring R, with the endomorphism algebra End ( M ) ( algebra of linear operators on M ) replacing the algebra of matrices, the analog of scalar matrices are scalar transformations.

For vector spaces, or more generally free modules, for which the endomorphism algebra is isomorphic to a matrix algebra, the scalar transforms are exactly the center of the endomorphism algebra, and similarly invertible transforms are the center of the general linear group GL ( V ), where they are denoted by Z ( V ), follow the usual notation for the center.

expansion in terms of the element Y ( and using the linear adjoint endomorphism notation, adX Y ≡ ), might serve well:

from to the Lie algebra of the general linear group GL ( V ), i. e. the endomorphism algebra of V.

where X and Y are tangent vector fields on M and s is a section of E. One must check that F < sup >∇</ sup > is C < sup >∞</ sup >- linear in both X and Y and that it does in fact define a bundle endomorphism of E.

In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V. A trivial representation of an associative or Lie algebra is a ( Lie ) algebra representation for which all elements of the algebra act as the zero linear map ( endomorphism ) which sends every element of V to the zero vector.

The linear transformation is also called the curvature transformation or endomorphism.

The most basic example is the Lie algebra of matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, This is the Lie algebra of the general linear group GL ( n ), and is reductive as it decomposes as corresponding to traceless matrices and scalar matrices.

If the algebra A is noncommutative, then the commutator with respect to an element of the algebra A defines a linear endomorphism of A to itself, which is a derivation over K. An algebra A equipped with a distinguished derivation d forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

A vertex algebra is a vector space V, together with an identity element 1 ∈ V, an endomorphism T: V → V, and a linear multiplication map

An element x of an affine algebraic group is unipotent when its associated right translation operator r < sub > x </ sub > on the affine coordinate ring A of G is locally unipotent as an element of the ring of linear endomorphism of A ( Locally unipotent means that its restriction to any finite dimensional stable subspace of A is unipotent in the usual ring sense ).

A linear endomorphism of this vector space is defined by counting the function's gradient flow lines connecting two critical points.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

From these results, one sees that the study of linear operators on vector spaces over an algebraically closed field is essentially reduced to the study of nilpotent operators.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

** On every infinite-dimensional topological vector space there is a discontinuous linear map.

His key contributions include topological tensor products of topological vector spaces, the theory of nuclear spaces as foundational for Schwartz distributions, and the application of L < sup > p </ sup > spaces in studying linear maps between topological vector spaces.

For a vector with linear addressing, the element with index i is located at the address B + c · i, where B is a fixed base address and c a fixed constant, sometimes called the address increment or stride.

where r is the position vector of the particle relative to the origin, p is the linear momentum of the particle, and × denotes the cross product.

In mathematics, a bilinear operator is a function combining elements of two vector spaces to yield an element of a third vector space that is linear in each of its arguments.

Note that if we regard the product as a vector space, then B is not a linear transformation of vector spaces ( unless or ) because, for example.

Two useful representations of a vector are simply a linear combination of basis vectors, and column matrices.

We are trying to study a linear operator T on the finite-dimensional space V, by decomposing T into a direct sum of operators which are in some sense elementary.

Analytical chemistry research is largely driven by performance ( sensitivity, selectivity, robustness, linear range, accuracy, precision, and speed ), and cost ( purchase, operation, training, time, and space ).

In general, a linear mapping on a normed space is continuous if and only if it is bounded on the closed unit ball.

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.