Linear algebra
For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations.
[4] Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry.
[5] In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra.
[6] The term vector was introduced as v = xi + yj + zk representing a point in space.
Linear algebra is flat differential geometry and serves in tangent spaces to manifolds.
The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modeling and simulations.
[10] Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps.
Gaussian elimination is the basic algorithm for finding these elementary operations, and proving these results.
To such a system, one may associate its matrix and its right member vector Let T be the linear transformation associated with the matrix M. A solution of the system (S) is a vector such that that is an element of the preimage of v by T. Let (S′) be the associated homogeneous system, where the right-hand sides of the equations are put to zero: The solutions of (S′) are exactly the elements of the kernel of T or, equivalently, M. The Gaussian-elimination consists of performing elementary row operations on the augmented matrix for putting it in reduced row echelon form.
Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear equations in n unknowns.
The Frobenius normal form does not need to extend the field of scalars and makes the characteristic polynomial immediately readable on the matrix.
The inner product is an example of a bilinear form, and it gives the vector space a geometric structure by allowing for the definition of length and angles.
Orthonormal bases are particularly easy to deal with, since if v = a1 v1 + ⋯ + an vn, then The inner product facilitates the construction of many useful concepts.
Thus, computing intersections of lines and planes amounts to solving systems of linear equations.
Until the end of the 19th century, geometric spaces were defined by axioms relating points, lines, and planes (synthetic geometry).
For improving efficiency, some of them configure the algorithms automatically, at run time, to adapt them to the specificities of the computer (cache size, number of available cores, ...).
Some contemporary processors, typically graphics processing units (GPU), are designed with a matrix structure, for optimizing the operations of linear algebra.
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.
To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting cells.
Linear models are frequently used for complex nonlinear real-world systems because they make parametrization more manageable.
[31] [32] [33] Linear algebra, a branch of mathematics dealing with vector spaces and linear mappings between these spaces, plays a critical role in various engineering disciplines, including fluid mechanics, fluid dynamics, and thermal energy systems.
For instance, linear algebraic techniques are used to solve systems of differential equations that describe fluid motion.
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.
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 grids.
It provides engineers with the necessary tools to model, analyze, and solve complex problems in these domains, leading to advancements in technology and industry.
One may thus replace the field of scalars by a ring R, and this gives the structure called a module over R, or R-module.
Functional analysis is of particular importance to quantum mechanics, the theory of partial differential equations, digital signal processing, and electrical engineering.
It also provides the foundation and theoretical framework that underlies the Fourier transform and related methods.
