# Abstract algebra

2017-07-27T17:51:21+03:00[Europe/Moscow] en true Direct limit, Algebraic number, Distributive property, Eigenvalues and eigenvectors, Lie algebra, Monoid, Ring (mathematics), Homomorphism, Isomorphism, Magma (algebra), Idempotence, Linear map, Transpose, Alternativity, Direct product, Scalar multiplication, Row and column spaces, Polarization identity, Dixmier conjecture flashcards Abstract algebra
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• Direct limit
In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family of objects".
• Algebraic number
An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
• Distributive property
In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from elementary algebra.
• Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that does not change its direction when that linear transformation is applied to it.
• Lie algebra
In mathematics, a Lie algebra (/liː/, not /laɪ/) is a vector space together with a non-associative multiplication called "Lie bracket" .
• Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
• Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
• Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces).
• Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse.
• Magma (algebra)
In abstract algebra, a magma (or groupoid; not to be confused with groupoids in category theory) is a basic kind of algebraic structure.
• Idempotence
Idempotence (/ˌaɪdᵻmˈpoʊtəns/ EYE-dəm-POH-təns) is the property of certain operations in mathematics and computer science, that can be applied multiple times without changing the result beyond the initial application.
• Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
• Transpose
In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr, tA or At) created by any one of the following equivalent actions: * reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT * write the rows of A as the columns of AT * write the columns of A as the rows of AT Formally, the i th row, j th column element of AT is the j th row, i th column element of A: If A is an m × n matrix then AT is an n × m matrix.
• Alternativity
In abstract algebra, alternativity is a property of a binary operation.
• Direct product
In mathematics, one can often define a direct product of objects already known, giving a new one.
• Scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).
• Row and column spaces
In linear algebra, the column space C(A) of a matrix A (sometimes called the range of a matrix) is the span (set of all possible linear combinations) of its column vectors.
• Polarization identity
In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.
• Dixmier conjecture
In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism.