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# 2 × 2 real matrices

In mathematics, the associative algebra of real matrices is denoted by M(2,&thinsp;R). 

53 relations: Abstract algebra, Addison-Wesley, Area, Associative algebra, Bijection, Change of basis, Commutative property, Complex analysis, Complex number, Connected space, D. Reidel, Determinant, Diagonal matrix, Differential form, Dot product, Dual number, Exterior algebra, General linear group, Hyperboloid, Hyperplane, Idempotent matrix, Identity component, Identity matrix, Imaginary unit, Invertible matrix, Involutory matrix, Linear map, Mathematics, Mathematics Magazine, Matrix (mathematics), Matrix addition, Matrix multiplication, Motor variable, Nilpotent matrix, Paraboloid, Polar decomposition, Projection (linear algebra), Rafael Artzy, Real line, Real number, Ring (mathematics), Ring homomorphism, Rotation (mathematics), Shear mapping, SL2(R), Special linear group, Split-complex number, Split-quaternion, Squeeze mapping, Subring, ... Expand index (3 more) »

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

Addison-Wesley is a publisher of textbooks and computer literature.

## Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

## Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

## Bijection

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.

## Change of basis

In linear algebra, a basis for a vector space of dimension n is a set of n vectors, called basis vectors, with the property that every vector in the space can be expressed as a unique linear combination of the basis vectors.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

## Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

## Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

## Dual number

In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2.

## Exterior algebra

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Hyperboloid

In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.

## Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

## Idempotent matrix

In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself.

## Identity component

In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group.

## Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

## Involutory matrix

In mathematics, an involutory matrix is a matrix that is its own inverse.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Mathematics Magazine

Mathematics Magazine is a refereed bimonthly publication of the Mathematical Association of America.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.

## Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

## Motor variable

In mathematics, a function of a motor variable is a function with arguments and values in the split-complex number plane, much as functions of a complex variable involve ordinary complex numbers.

## Nilpotent matrix

In linear algebra, a nilpotent matrix is a square matrix N such that for some positive integer k. The smallest such k is sometimes called the index of N. More generally, a nilpotent transformation is a linear transformation L of a vector space such that Lk.

## Paraboloid

In geometry, a paraboloid is a quadric surface that has (exactly) one axis of symmetry and no center of symmetry.

## Polar decomposition

In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to the polar form of a nonzero complex number z as z.

## Projection (linear algebra)

In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

## Rafael Artzy

Rafael Artzy (23 July 1912 – 22 August 2006) was an Israeli mathematician specializing in geometry.

## Real line

In mathematics, the real line, or real number line is the line whose points are the real numbers.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.

## Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

## Shear mapping

In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.

## SL2(R)

In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \right): a,b,c,d\in\mathbf\mboxad-bc.

## Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

## Split-complex number

In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.

## Split-quaternion

In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name.

## Squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

## Subring

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).

## Unit (ring theory)

In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.

## University of Chicago Press

The University of Chicago Press is the largest and one of the oldest university presses in the United States.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## References

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