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algebra over a field | A Wisdom Archive on algebra over a field | | algebra over a field A selection of articles related to algebra over a field | |
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More material related to Algebra Over A Field can be found here:
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| | | algebra over a field, Algebra over a field - Definitions, Algebra over a field - Index-free notation, Algebra over a field - K-algebra morphism, Algebra over a field - Kinds of algebras and examples, Algebra over a field - Properties, Clifford algebra, Geometric algebra | | | » Page 1 « Page 2 | |
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| ARTICLES RELATED TO algebra over a field | |
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| | |  algebra over a field: Encyclopedia II - Vector space - Formal definitionA vector space over a field F (such as the field of real numbers or the field of complex numbers) is a set V together with two operations,
vector addition, denoted v + w, where v, w ∈ V, mapping V × V (see Cartesian product) back into V, and
scalar multiplication, denoted a v, where a ∈ F and v ∈ V, ...
See also:Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures Read more here: » Vector space: Encyclopedia II - Vector space - Formal definition |
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| | | | algebra over a field: Encyclopedia II - Polynomial ring - The polynomial ring in several variables Given two variables X and Y, one constructs the polynomial ring R[X], and then, on top of it, the ring (R[X])[Y]. This ring is considered the polynomial ring in the two variables R[X,Y].
For example, the polynomial
P(X,Y) = X2Y2 + 4XY2 + 5X3 â ...
See also:Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings Read more here: » Polynomial ring: Encyclopedia II - Polynomial ring - The polynomial ring in several variables |
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| | | | algebra over a field: Encyclopedia II - Lie algebra - Relation to Lie groups Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups. Given a Lie group, a Lie algebra can be associated to it either by endowing the tangent space to the identity with the differential of the adjoint map, or by considering the left-invariant vector fields as mentioned in the examples. This association is functorial, meaning that homomorphisms of Lie groups lift to homomorphisms of Lie algebras, and various properties are satisfied by this lifting: it commutes with composition, it maps subgroups, kernels, quotients and cokernels of Lie groups to subalgebras, kernels, ...
See also:Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Relation to Lie groups |
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| | | | algebra over a field: Encyclopedia II - Vector space - Formal definition A vector space over a field F (such as the field of real or of complex numbers) is a set V together with two operations:
vector addition: defined on the Cartesian product V × V with values in V and denoted v + w, where v, w ∈ V, and
scalar multiplication: defined on the Cartesian product F × V with values in V and denoted a v, where a< ...
See also:Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures Read more here: » Vector space: Encyclopedia II - Vector space - Formal definition |
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| | | | algebra over a field: Encyclopedia II - Polynomial ring - The polynomial ring R[X] One can then check that the set of all polynomials with coefficients in the ring R, together with the addition + and the multiplication mentioned above, forms itself a ring, the polynomial ring over R, which is denoted by R[X].
Formally these two ring operations are functions defined on with values in R[X], given by the formulas
and
If ...
See also:Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings Read more here: » Polynomial ring: Encyclopedia II - Polynomial ring - The polynomial ring R[X] |
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| | | | algebra over a field: Encyclopedia II - Lie algebra - Homomorphisms subalgebras and ideals A homomorphism φ : g → h between Lie algebras g and h over the same base field F is an F-linear map such that [φ(x), φ(y)] = φ([x, y]) for all x and y in g. The composition of such homomorphisms is again a homomorphism, and the Lie algebras over the field F, together with these morphisms, form a category. If such a homomorphism is bijective, it is called an isomorphism, and the two Lie algebras g and h are called isomorphic ...
See also:Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Homomorphisms subalgebras and ideals |
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| | | | algebra over a field: Encyclopedia II - Lie algebra - Classification of Lie algebras Real and complex Lie algebras can be classified to some extent, and this classification is an important step toward the classification of Lie groups. Every finite-dimensional real or complex Lie algebra arises as the Lie algebra of unique real or complex simply connected Lie group (Ado's theorem), but there may be more than one group, even more than one connected group, giving rise to the same algebra. For instance, the groups SO(3) (3×3 orthogonal matrices of determinant 1) and SU(2) (2×2 unitary matrices of determinant 1) both give rise to the sam ...
See also:Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Classification of Lie algebras |
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| | | | algebra over a field: Encyclopedia II - Polynomial ring - Definition of a polynomial In real analysis, a polynomial is a certain type of a function of one or several variables (see polynomial), or in other words, a polynomial function.
This definition cannot be adapted to a general ring, however. For example, over the ring Z/2Z of integers modulo 2, the polynomial
P(X)=X2+X=X(X+1)
takes only the value 0, as when k is an integer, k(k+1) is always even. But we would expec ...
See also:Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings Read more here: » Polynomial ring: Encyclopedia II - Polynomial ring - Definition of a polynomial |
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| | | | algebra over a field: Encyclopedia II - Lie algebra - Definition A Lie algebra is a type of an algebra over a field; it is a vector space g over some field F together with a binary operation [·, ·] : g × g → g, called the Lie bracket, which satisfies the following properties:
Bilinearity:
for all a, b ∈ F and all x, y, z ∈ g.
For all x in g
< ...
See also:Lie algebra, Lie algebra - Definition, Lie algebra - Examples, Lie algebra - Homomorphisms subalgebras and ideals, Lie algebra - Relation to Lie groups, Lie algebra - Classification of Lie algebras, Lie algebra - Category theoretic definition Read more here: » Lie algebra: Encyclopedia II - Lie algebra - Definition |
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| | | | algebra over a field: Encyclopedia II - Vector space - Generalizations and additional structures It is common to study vector spaces with certain additional structures. This is often necessary for recovering ordinary notions from geometry. Some of these additional structures include:
A real or complex vector space with a defined length concept, i.e., a norm, is called a normed vector space.
A real or complex vector space with a notion of both length and angle is called an inner product space.
A vector space with a topology compatible with the operations (i.e., such that addition and scalar mu ...
See also:Vector space, Vector space - Formal definition, Vector space - Elementary properties, Vector space - Examples, Vector space - Subspaces and bases, Vector space - Linear transformations, Vector space - Generalizations and additional structures Read more here: » Vector space: Encyclopedia II - Vector space - Generalizations and additional structures |
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