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.

** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals

* Jansky radio astronomer's preferred unit – linear in power / unit area

Scanning velocity is 1. 2 – 1. 4 m / s ( constant linear velocity ) – equivalent to approximately 500 rpm at the inside of the disc, and approximately 200 rpm at the outside edge.

" From these principles and some additional constraints —( 1a ) a lower bound on the linear dimensions of any of the parts, ( 1b ) an upper bound on speed of propagation ( the velocity of light ), ( 2 ) discrete progress of the machine, and ( 3 ) deterministic behavior — he produces a theorem that " What can be calculated by a device satisfying principles I – IV is computable.

* Diffraction approximations illustrated – MIT site that illustrates the various approximations in diffraction and intuitively explains the Fraunhofer regime from the perspective of linear system theory.

In the hyperbolic case the Hartman – Grobman theorem gives the conditions for the existence of a continuous function that maps the neighborhood of the fixed point of the map to the linear map J · x.

The Hahn – Banach theorem states that if is a sublinear function, and is a linear functional on a linear subspace U ⊆ V which is dominated by on U,

For molecules with N atoms in them, linear molecules have 3N – 5 degrees of vibrational modes, whereas nonlinear molecules have 3N – 6 degrees of vibrational modes ( also called vibrational degrees of freedom ).

More advanced statistical methods employed by some I – O psychologists include logistic regression, multivariate analysis of variance, structural equation modeling, and hierarchical linear modeling ( HLM ; also known as multilevel modeling ).

Second is the mass – luminosity relation, which relates the luminosity L and the mass M. Finally, the relationship between M and R is close to linear.

* Rectilinear motion ( Linear motion ) – motion which follows a straight linear path, and whose displacement is exactly the same as its trajectory.

Amylose consists of a linear chain of several hundred glucose molecules and Amylopectin is a branched molecule made of several thousand glucose units ( every chain of 24 – 30 glucose units is one unit of Amylopectin ).

This is elementary linear algebra – one direction is trivial ; the other follows from:

Note that the power injected by the torque depends only on the instantaneous angular speed – not on whether the angular speed increases, decreases, or remains constant while the torque is being applied ( this is equivalent to the linear case where the power injected by a force depends only on the instantaneous speed – not on the resulting acceleration, if any ).

In colloquial usage, the terms " Turing complete " or " Turing equivalent " are used to mean that any real-world general-purpose computer or computer language can approximately simulate any other real-world general-purpose computer or computer language, within the bounds of finite memory – they are linear bounded automaton complete.

* Algebraic code-excited linear prediction ( ACELP 4. 7 kbit / s – 24 kbit / s )

The concept of wavelength is most often applied to sinusoidal, or nearly sinusoidal, waves, because in a linear system the sinusoid is the unique shape that propagates with no shape change – just a phase change and potentially an amplitude change.

Due to the Kramers – Kronig relations, a linear medium with dispersion also exhibits loss, so the sine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon the medium.

The wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves.

Take as an example a program that looks up a specific entry in a sorted list of size n. Suppose this program were implemented on Computer A, a state-of-the-art machine, using a linear search algorithm, and on Computer B, a much slower machine, using a binary search algorithm.

Modern methods include the use of lossless data compression for incremental parsing, prediction suffix tree and string searching by factor oracle algorithm ( basically a factor oracle is a finite state automaton constructed in linear time and space in an incremental fashion ).

* Conjugate gradient method, an algorithm for the numerical solution of particular systems of linear equations

Another theoretical attack, linear cryptanalysis, was published in 1994, but it was a brute force attack in 1998 that demonstrated that DES could be attacked very practically, and highlighted the need for a replacement algorithm.

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.

In 1970 Danny Cohen presented at the " Computer Graphics 1970 " conference in England a linear algorithm for drawing ellipses and circles.

A 4-way floodfill algorithm that use the adjacency technique and a stack as its seed pixel store yields a linear fill with " gaps filled later " behaviour.

showed that a related problem ( EUGCD, determining the remainder sequence arising during the Euclidean algorithm ) is NC-equivalent to the problem of integer linear programming with two variables ; if either problem is in NC or is P-complete, the other is as well.

In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations.

This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete.

Depending on the algorithm used, other properties may be required as well, such as double hashing and linear probing.

* Merge sort parallelizes well and can achieve close to linear speedup with a trivial implementation ; heapsort is not an obvious candidate for a parallel algorithm.

The most widely used learning algorithms are Support Vector Machines, linear regression, logistic regression, naive Bayes, linear discriminant analysis, decision trees, k-nearest neighbor algorithm, and Neural Networks ( Multilayer perceptron ).

There are also search methods designed for quantum computers, like Grover's algorithm, that are theoretically faster than linear or brute-force search even without the help of data structures or heuristics.

Describes a randomized half-plane intersection algorithm for linear programming.

Provided that the edges are either already sorted or can be sorted in linear time ( for example with counting sort or radix sort ), the algorithm can use more sophisticated disjoint-set data structure to run in O ( E α ( V )) time, where α is the extremely slowly growing inverse of the single-valued Ackermann function.

On the other hand, if T is written as a unary number ( a string of n ones, where n = T ), then it only takes time n. By writing T in unary rather than binary, we have reduced the obvious sequential algorithm from exponential time to linear time.

Shamir has also made contributions to computer science outside of cryptography, such as finding the first linear time algorithm for 2-satisfiability and showing the equivalence of the complexity classes PSPACE and IP.

Notice that during the entire computation, the state of the algorithm is a linear combination of and.

For this reason, methods based on introductory linear algebra texts are generally not suitable for implementation in software ; rather, one should consult contemporary numerical analysis sources for an algorithm like the one below, which does not amplify rounding errors unnecessarily.

The original optimal hyperplane algorithm proposed by Vapnik in 1963 was a linear classifier.

Using big-O notation, the performance of the interpolation algorithm on a data set of size N is O ( N ); however under the assumption of a uniform distribution of the data on the linear scale used for interpolation, the performance can be shown to be O ( log log N ).