Much of the mathematics now taught to engineers and physicists requires it. This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related topics such as determinants, eigenvalues, and differential equations.
Table of Contents: l. The Algebra of Matrices 2. Linear Equations 3. Vector Spaces 4.
Determinants 5. Linear Transformations 6. This requires the heavy use of linear algebra. Functional analysis studies function spaces. These are vector spaces with additional structure, such as Hilbert spaces. Linear algebra is thus a fundamental part of functional analysis and its applications, which include, in particular, quantum mechanics wave functions. Most physical phenomena are modeled by partial differential equations.
To solve them, one usually decomposes the space in which the solutions are searched into small, mutually interacting cells. For linear systems this interaction involves linear functions. For nonlinear systems , this interaction is often approximated by linear functions.
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Nearly all scientific computations involve linear algebra. Consequently, linear algebra algorithms have been highly optimized. For improving efficiency, some of them configure the algorithms automatically, at run time, for adapting them to the specificities of the computer cache size, number of available cores , Some processors , typically graphics processing units GPU , are designed with a matrix structure, for optimizing the operations of linear algebra.
This section presents several related topics that do not appear generally in elementary textbooks on linear algebra, but are commonly considered, in advanced mathematics, as parts of linear algebra. The existence of multiplicative inverses in fields is not involved in the axioms defining a vector space. One may thus replace the field of scalars by a ring R , and this gives a structure called module over R , or R -module.
The concepts of linear independence, span, basis, and linear maps also called module homomorphisms are defined for modules exactly as for vector spaces, with the essential difference that, if R is not a field, there are modules that do not have any basis. The modules that have a basis are the free modules , and those that are spanned by a finite set are the finitely generated modules. Module homomorphisms between finitely generated free modules may be represented by matrices. The theory of matrices over a ring is similar to that of matrices over a field, except that determinants exist only if the ring is commutative , and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring.
Vector spaces are completely characterized by their dimension up to an isomorphism. In general, there is not such a complete classification for modules, even if one restricts oneself to finitely generated modules. However, every module is a cokernel of a homomorphism of free modules. Modules over the integers can be identified with abelian groups , since the multiplication by an integer may identified to a repeated addition. Most of the theory of abelian groups may be extended to modules over a principal ideal domain.
In particular, over a principal ideal domain, every submodule of a free module is free, and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a principal ring. There are many rings for which there are algorithms for solving linear equations and systems of linear equations. However, these algorithms have generally a computational complexity that is much higher than the similar algorithms over a field.
For more details, see Linear equation over a ring. In multilinear algebra , one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. Vector spaces that are not finite dimensional often require additional structure to be tractable. A normed vector space is a vector space along with a function called a norm , which measures the "size" of elements.
The norm induces a metric , which measures the distance between elements, and induces a topology , which allows for a definition of continuous maps. The metric also allows for a definition of limits and completeness - a metric space that is complete is known as a Banach space. A complete metric space along with the additional structure of an inner product a conjugate symmetric sesquilinear form is known as a Hilbert space , which is in some sense a particularly well-behaved Banach space. Functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces; the central objects of study in functional analysis are L p spaces , which are Banach spaces, and especially the L 2 space of square integrable functions, which is the only Hilbert space among them.
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. From Wikipedia, the free encyclopedia. Branch of mathematics. Main article: Vector space. Main article: Linear map. Main articles: Linear subspace , Linear span , and Basis linear algebra.
Main article: Matrix mathematics. Main article: System of linear equations. Main article: Square matrix. Main article: Determinant. Main article: Eigenvalues and eigenvectors. Main article: Dual space. Main article: Transpose of a linear map. This section may require cleanup to meet Wikipedia's quality standards. The specific problem is: Need for a more encyclopedic style, which is homogeneous with the style of preceding sections.
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Also, some details do not belong to this general article but to more specialized ones. Also, inner product spaces should appear as a special instance of the more general concept of bilinear form. Finally, complex conjugation should appear in a specific section on linear algebra over the complexes. Please help improve this section if you can. August Learn how and when to remove this template message. Main article: Inner product space.
Main article: Module mathematics. The specific problem is: The dual space is considered above, and the section must be rewritten for given a understandable summary of this subject Please help improve this section if you can. September Learn how and when to remove this template message. This section needs expansion. You can help by adding to it. September Main articles: Topological vector space , Normed vector space , and Hilbert space.
Main article: Homological algebra. Retrieved 16 April The Chinese Roots of Linear Algebra. JHU Press. Department of Mathematics. University of Oregon. Archived from the original on Retrieved Van Nostrand, , Google Eprint.
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Jain, Khalil Ahmad Functional analysis 2nd ed. New Age International. Quantum mechanics in Hilbert space 2nd ed. Academic Press.
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Linear algebra. Linear Algebra. Georgi E. Fourier Analysis. Rami Shakarchi. Advanced Calculus Problem Solver. Editors of REA.