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 linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL ( V ).

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.

Specifically, in quantum mechanics, the state of an atom, i. e. an eigenstate of the atomic Hamiltonian, is approximated by an expansion ( see configuration interaction expansion and basis set ) into linear combinations of anti-symmetrized products ( Slater determinants ) of one-electron functions.

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

The vector A is still conventionally represented by a linear combination of basis vectors or a column matrix:

Since these are a basis, any quantum state of the particle can be expressed as a linear combination ( i. e., quantum superposition ) of these two states:

In linear algebra, if two endomorphisms of a space are represented by commuting matrices with respect to one basis, then they are so represented with respect to every basis.

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

Families in this group have been assigned on the basis of the nature of the genome ( circular or linear ) and the host range.

The transform is sometimes denoted by the symbol, as in or or < ref group =" note "> As a linear transformation on a finite-dimensional vector space, the DFT expression can also be written in terms of a DFT matrix ; when scaled appropriately it becomes a unitary matrix and the X < sub > k </ sub > can thus be viewed as coefficients of x in an orthonormal basis .</ ref >

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

A classic application for a pair of class-A devices is the long-tailed pair, which is exceptionally linear, and forms the basis of many more complex circuits, including many audio amplifiers and almost all op-amps.

This echelon matrix contains a wealth of information about: the rank of is 5 since there are 5 non-zero rows in ; the vector space spanned by the columns of has a basis consisting of the first, third, fourth, seventh and ninth column of ( the columns of the ones in ), and the *' s tell you how the other columns of can be written as linear combinations of the basis columns.

Saturated hydrocarbons are the basis of petroleum fuels and are found as either linear or branched species.

Given a basis of a vector space, every element of the vector space can be expressed uniquely as a finite linear combination of basis vectors.

* The number of molecular orbitals is equal to the number of atomic orbitals included in the linear expansion or the basis set

composed of a linear combination of basis B-splines b < sub > i, n </ sub > of degree n

The model assumes that there are two “ equally positioned firms ”; the firms compete on the basis of quantity rather than price and each firm makes an “ output decision assuming that the other firm ’ s behavior is fixed .” The market demand curve is assumed to be linear and marginal costs are constant.

A pure qubit state is a linear superposition of the basis states.

* Informally, by extension, the line generated by a basis vector in linear algebra

When this linear draft is completed, I dust it down to a faint image.

The theory predicts a linear dependence of Af on Af, where J is the experimentally determined Curie-Weiss constant.

If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;

If Af are the projections associated with the primary decomposition of T, then each Af is a polynomial in T, and accordingly if a linear operator U commutes with T then U commutes with each of the Af, i.e., each subspace Af is invariant under U.

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.

that is, one must know something about D other than the fact that it is a linear operator.

UTC is a discontinuous time scale composed from segments that are linear transformations of atomic time, the discontinuities being arranged so that UTC approximated UT2 until the end of 1971, and UT1 thereafter.

# As exploratory data analysis, an ANOVA is an organization of an additive data decomposition, and its sums of squares indicate the variance of each component of the decomposition ( or, equivalently, each set of terms of a linear model ).

# Closely related to the ANOVA is a linear model fit with coefficient estimates and standard errors.

Even when the statistical model is nonlinear, it can be approximated by a linear model for which an analysis of variance may be appropriate.

Since the randomization-based analysis is complicated and is closely approximated by the approach using a normal linear model, most teachers emphasize the normal linear model approach.

ANOVA is considered to be a special case of linear regression

which in turn is a special case of the general linear model.

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