Are You a Linear Algebra Pro?

Linear Algebra: A Summary

Linear algebra deals with linear equations, linear maps, vector spaces, and matrices.
It is a fundamental topic in modern presentations of geometry and is used in almost all areas of mathematics, science, and engineering.
Linear algebra is used to model natural phenomena and to efficiently compute with such models.
The procedure for solving simultaneous linear equations using counting rods appears in ancient Chinese mathematical texts.
Systems of linear equations arose in Europe with the introduction of coordinates in geometry by René Descartes in 1637.
The first systematic methods for solving linear systems used determinants and were first considered by Leibniz in 1693.
Linear algebra grew with ideas noted in the complex plane and the discovery of the four-dimensional system of quaternions by W.R. Hamilton in 1843.
Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group.
Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra.
A vector space over a field F is a set V equipped with two binary operations: vector addition and scalar multiplication.
Linear maps are mappings between vector spaces that preserve the vector-space structure.
Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps, and their theory is thus an essential part of linear algebra.Linear Algebra: Key Concepts and Applications
Linear algebra involves the study of linear equations, vector spaces, matrices, and linear transformations.
Elementary row operations are used to put the augmented matrix of a linear system in reduced row echelon form, which helps in finding the unique solution to the system.
Linear endomorphisms, represented by square matrices, are important in many areas of mathematics, including geometric transformations and coordinate changes.
The determinant of a square matrix is used to determine if the matrix is invertible, and Cramer's rule is an expression for the solution of a system of linear equations in terms of determinants.
Eigenvalues and eigenvectors are important concepts in linear algebra, and diagonalizable matrices have a very simple structure.
Duality is a concept in linear algebra that involves the dual space of a vector space and the dual or transpose of a linear map.
Inner product spaces, which give vector spaces a geometric structure, are studied in linear algebra and facilitate the construction of many useful concepts.
Linear algebra is widely used in geometry, mechanics, robotics, computer vision, computer graphics, and many other scientific domains.
Functional analysis studies function spaces, which are vector spaces with additional structure, and linear algebra is a fundamental part of functional analysis and its applications.
Linear algebra is used for solving partial differential equations and for modeling complex nonlinear systems.
Linear algebra algorithms have been highly optimized for scientific computation, and some processors, such as GPUs, are designed with a matrix structure.
Module theory is a generalization of vector spaces that involves modules over rings, and it has applications in group theory and number theory.Overview of Vector Spaces and Related Topics
Algorithms for solving linear equations over a ring are generally more complex than those for solving equations over a field.
Multilinear algebra involves multivariable linear transformations and leads to the concept of dual spaces and tensor products.
A vector space with a bilinear vector product is called an algebra, such as the algebra of square matrices or polynomials.
Normed vector spaces have a norm function that measures the “size” of elements and induces a metric and topology for the space.
Functional analysis applies linear algebra and mathematical analysis to function spaces, with Lp spaces and L2 space being important examples.
Homological algebra is another area of study related to vector spaces.
The Fourier transform and related methods are based on functional analysis.
Quantum mechanics, partial differential equations, digital signal processing, and electrical engineering all rely on functional analysis.
Normed vector spaces that are complete are known as Banach spaces.
A complete metric space with an inner product is a Hilbert space.
Topological vector spaces are necessary for tractable vector spaces that are not finite dimensional.
Associative algebras have an associative vector product.
The algebra of polynomials is an example of an algebra.
Multilinear maps can be described via tensor products of elements of V*.
Dual spaces consist of linear maps f : V → F.
Multilinear algebra involves mappings that are linear in multiple variables.