Katana VentraIP

Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as:

linear maps such as:


and their representations in vector spaces and through matrices.[1][2][3]


Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to function spaces.


Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.

symmetry:

Conjugate

Relationship with geometry[edit]

There is a strong relationship between linear algebra and geometry, which started with the introduction by René Descartes, in 1637, of Cartesian coordinates. In this new (at that time) geometry, now called Cartesian geometry, points are represented by Cartesian coordinates, which are sequences of three real numbers (in the case of the usual three-dimensional space). The basic objects of geometry, which are lines and planes are represented by linear equations. Thus, computing intersections of lines and planes amounts to solving systems of linear equations. This was one of the main motivations for developing linear algebra.


Most geometric transformation, such as translations, rotations, reflections, rigid motions, isometries, and projections transform lines into lines. It follows that they can be defined, specified and studied in terms of linear maps. This is also the case of homographies and Möbius transformations, when considered as transformations of a projective space.


Until the end of the 19th century, geometric spaces were defined by axioms relating points, lines and planes (synthetic geometry). Around this date, it appeared that one may also define geometric spaces by constructions involving vector spaces (see, for example, Projective space and Affine space). It has been shown that the two approaches are essentially equivalent.[22] In classical geometry, the involved vector spaces are vector spaces over the reals, but the constructions may be extended to vector spaces over any field, allowing considering geometry over arbitrary fields, including finite fields.


Presently, most textbooks, introduce geometric spaces from linear algebra, and geometry is often presented, at elementary level, as a subfield of linear algebra.

Fundamental matrix (computer vision)

Geometric algebra

Linear programming

a statistical estimation method

Linear regression

Numerical linear algebra

Outline of linear algebra

Transformation matrix

Fearnley-Sander, Desmond, "", American Mathematical Monthly 86 (1979), pp. 809–817.

Hermann Grassmann and the Creation of Linear Algebra

(1844), Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, Leipzig: O. Wigand

Grassmann, Hermann

a series of 34 recorded lectures by Professor Gilbert Strang (Spring 2010)

MIT Linear Algebra Video Lectures

International Linear Algebra Society

, Encyclopedia of Mathematics, EMS Press, 2001 [1994]

"Linear algebra"

on MathWorld

Linear Algebra

on Earliest Known Uses of Some of the Words of Mathematics

Matrix and Linear Algebra Terms

on Earliest Uses of Various Mathematical Symbols

Earliest Uses of Symbols for Matrices and Vectors

a video presentation from 3Blue1Brown of the basics of linear algebra, with emphasis on the relationship between the geometric, the matrix and the abstract points of view

Essence of linear algebra