Editing Linear algebra (section)

== Usage and applications{{anchor|Applications}} ==
Linear algebra is used in almost all areas of mathematics, thus making it relevant in almost all scientific domains that use mathematics. These applications may be divided into several wide categories.

=== Functional analysis ===
[[Functional analysis]] studies [[function space]]s. These are vector spaces with additional structure, such as [[Hilbert space]]s. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, [[quantum mechanics]] ([[wave function]]s) and [[Fourier analysis]] ([[orthogonal basis]]).

=== Scientific computation ===
Nearly all [[scientific computation]]s involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. [[Basic Linear Algebra Subprograms|BLAS]] and [[LAPACK]] are the best known implementations. For improving efficiency, some of them configure the algorithms automatically, at run time, to adapt them to the specificities of the computer ([[cache (computing)|cache]] size, number of available [[multi-core processor|cores]],&nbsp;...).

Since the 1960s there have been processors with specialized instructions<ref>{{cite book
 | title     = IBM System/36O Model 40 - Sum of Products Instruction-RPQ W12561 - Special Systems Feature
 | id        = L22-6902
 | publisher = [[IBM]]
 }}
</ref> for optimizing the operations of linear algebra, optional array processors<ref>{{cite book
 | title     = IBM System/360 Custom Feature Description: 2938 Array Processor Model 1, - RPQ W24563; Model 2, RPQ 815188
 | id        = A24-3519
 | publisher = [[IBM]]
 }}
</ref> under the control of a conventional processor, supercomputers<ref>{{cite journal
 | journal     = [[IEEE Transactions on Computers]]
 | title       = The ILLIAC IV Computer
 | last1       = Barnes
 | first1      = George
 | last2       = Brown
 | first2      = Richard
 | last3       = Kato
 | first3      = Maso
 | last4       = Kuck
 | first4      = David
 | last5       = Slotnick
 | first5      = Daniel
 | last6       = Stokes
 | first6      = Richard
 | date        = August 1968
 | volume      = C.17
 | number      = 8
 | pages       = 746–757
 | issn        = 0018-9340 
 | s2cid       = 206617237
 | doi         = 10.1109/tc.1968.229158
 | url         = http://gordonbell.azurewebsites.net/cgb%20files/computer%20structures%20readings%20and%20examples%201971.pdf
 | access-date = October 31, 2024
 }}
</ref><ref name="Star100HW">{{cite book
 | title       = Star-100 - Hardware Reference Manual
 | id          = 60256000
 | date        = December 15, 1975
 | version     = Revision 9
 | publisher   = [[Control Data Corporation]]
 | url         = http://bitsavers.trailing-edge.com/pdf/cdc/cyber/cyber_200/60256000_STAR-100hw_Dec75.pdf
 | access-date = October 31, 2024
 }}
</ref><ref name="cray1hw">{{cite book
 | title       = Cray-1 - Computer System - Hardware Reference Manual
 | id          = 2240004
 | date        = November 4, 1977
 | version     = Rev. C
 | publisher   = [[Cray Research, Inc.]]
 | url         = http://bitsavers.trailing-edge.com/pdf/cray/CRAY-1/2240004C_CRAY-1_Hardware_Reference_Nov77.pdf
 | access-date = October 31, 2024
 }}
</ref> designed for array processing and conventional processors augmented<ref>{{cite book
 | title       = IBM Enterprise Systems Architecture/370 and System/370 Vector Operations 
 | id          = SA22-7125-3
 | date        = August 1988
 | edition     = Fourth 
 | publisher   = [[IBM]]
 | url         = http://bitsavers.org/pdf/ibm/370/vectorFacility/SA22-7125-3_Vector_Operations_Aug88.pdf
 | access-date = October 31, 2024
 }}
</ref> with vector registers.

Some contemporary [[Processor (computing)|processor]]s, typically [[graphics processing units]] (GPU), are designed with a matrix structure, for optimizing the operations of linear algebra.<ref>{{Cite web |title=GPU Performance Background User's Guide |url=https://docs.nvidia.com/deeplearning/performance/dl-performance-gpu-background/index.html |access-date=2024-10-29 |website=NVIDIA Docs |language=en}}</ref>

=== Geometry of ambient space ===
The [[Mathematical model|modeling]] of [[ambient space]] is based on [[geometry]]. Sciences concerned with this space use geometry widely. This is the case with [[mechanics]] and [[robotics]], for describing [[rigid body dynamics]]; [[geodesy]] for describing [[Earth shape]]; [[perspectivity]], [[computer vision]], and [[computer graphics]], for describing the relationship between a scene and its plane representation; and many other scientific domains.

In all these applications, [[synthetic geometry]] is often used for general descriptions and a qualitative approach, but for the study of explicit situations, one must compute with [[coordinates]]. This requires the heavy use of linear algebra.

=== Study of complex systems ===
{{see also|Complex system}}
Most physical phenomena are modeled by [[partial differential equation]]s. To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting [[Discretization|cells]]. For [[linear system]]s this interaction involves [[linear function]]s. For [[nonlinear systems]], this interaction is often approximated by linear functions.{{efn|This may have the consequence that some physically interesting solutions are omitted.}}This is called a linear model or first-order approximation. Linear models are frequently used for complex nonlinear real-world systems because they make [[Parametrization (geometry)|parametrization]] more manageable.<ref>{{Cite book |last=Savov |first=Ivan |title=No Bullshit Guide to Linear Algebra |publisher=MinireferenceCo. |year=2017 |isbn=9780992001025 |pages=150–155 |language=en}}</ref> In both cases, very large matrices are generally involved. [[Weather forecasting]] (or more specifically, [[Parametrization (atmospheric modeling)|parametrization for atmospheric modeling]]) is a typical example of a real-world application, where the whole Earth [[atmosphere]] is divided into cells of, say, 100&nbsp;km of width and 100&nbsp;km of height.

=== Fluid mechanics, fluid dynamics, and thermal energy systems ===
<ref>{{Cite web|title= MIT OpenCourseWare. Special Topics in Mathematics with Applications: Linear Algebra and the Calculus of Variations - Mechanical Engineering |url= https://ocw.mit.edu/courses/2-035-special-topics-in-mathematics-with-applications-linear-algebra-and-the-calculus-of-variations-spring-2007/}}</ref><ref>{{Cite web|title= FAMU-FSU College of Engineering. ME Undergraduate Curriculum |url= https://engineering.ucdenver.edu/electrical-engineering/research/energy-and-power-systems#:~:text=Power%20systems%20analysis%20deals%20with,the%20analysis%20of%20power%20systems}}</ref><ref>{{Cite web|title= University of Colorado Denver. Energy and Power Systems |url= https://eng.famu.fsu.edu/me/undergraduate-curriculum#:~:text=MAS%203105%20Linear%20Algebra%20%283%29,and%20eigenvectors%2C%20linear%20transformations%2C%20applications)}}</ref>

Linear algebra, a branch of mathematics dealing with [[vector spaces]] and [[linear mapping]]s between these spaces, plays a critical role in various engineering disciplines, including [[fluid mechanics]], [[fluid dynamics]], and [[thermal energy]] systems. Its application in these fields is multifaceted and indispensable for solving complex problems.

In [[fluid mechanics]], linear algebra is integral to understanding and solving problems related to the behavior of fluids. It assists in the modeling and simulation of fluid flow, providing essential tools for the analysis of [[fluid dynamics]] problems. For instance, linear algebraic techniques are used to solve systems of [[differential equations]] that describe fluid motion. These equations, often complex and [[non-linear]], can be linearized using linear algebra methods, allowing for simpler solutions and analyses.

In the field of fluid dynamics, linear algebra finds its application in [[computational fluid dynamics]] (CFD), a branch that uses [[numerical analysis]] and [[data structure]]s to solve and analyze problems involving fluid flows. CFD relies heavily on linear algebra for the computation of fluid flow and [[heat transfer]] in various applications. For example, the [[Navier–Stokes equations]], fundamental in [[fluid dynamics]], are often solved using techniques derived from linear algebra. This includes the use of [[Matrix (mathematics)|matrices]] and [[Vector (mathematics and physics)|vectors]] to represent and manipulate fluid flow fields.

Furthermore, linear algebra plays a crucial role in [[thermal energy]] systems, particularly in [[power systems]] analysis. It is used to model and optimize the generation, [[Electric power transmission|transmission]], and [[Electric power distribution|distribution]] of electric power. Linear algebraic concepts such as matrix operations and [[eigenvalue]] problems are employed to enhance the efficiency, reliability, and economic performance of [[power systems]]. The application of linear algebra in this context is vital for the design and operation of modern [[power systems]], including [[renewable energy]] sources and [[smart grid]]s.

Overall, the application of linear algebra in [[fluid mechanics]], [[fluid dynamics]], and [[thermal energy]] systems is an example of the profound interconnection between [[mathematics]] and [[engineering]]. It provides engineers with the necessary tools to model, analyze, and solve complex problems in these domains, leading to advancements in technology and industry.