Search results
Results from the WOW.Com Content Network
for v, v 1, v 2 ∈ V, w, w 1, w 2 ∈ W, and α ∈ K. The resulting vector space is called the direct sum of V and W and is usually denoted by a plus symbol inside a circle: It is customary to write the elements of an ordered sum not as ordered pairs (v, w), but as a sum v + w. The subspace V × {0} of V ⊕ W is isomorphic to V and is often ...
An element in the direct product is an infinite sequence, such as (1,2,3,...) but in the direct sum, there is a requirement that all but finitely many coordinates be zero, so the sequence (1,2,3,...) would be an element of the direct product but not of the direct sum, while (1,2,0,0,0,...) would be an element of both.
This applies also when E and F are linear subspaces or submodules of the vector space or module V. 2. Direct sum: if E and F are two abelian groups, vector spaces, or modules, then their direct sum, denoted is an abelian group, vector space, or module (respectively) equipped with two monomorphisms: and : such that is the internal direct sum of ...
Formally, the symmetric algebra of a vector space V over a field F is the group algebra of the dual, Sym(V) := F[V ∗], and the Weyl algebra is the group algebra of the (dual) Heisenberg group W(V) = F[H(V ∗)]. Since passing to group algebras is a contravariant functor, the central extension map H(V) → V becomes an inclusion Sym(V) → W(V).
Let T : V → W be a linear operator. The kernel of T, denoted ker(T), is the set of all x in V such that Tx = 0. The kernel is a subspace of V. The first isomorphism theorem for vector spaces says that the quotient space V/ker(T) is isomorphic to the image of V in W.
For example, + + is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x 3 + 3 x 2 y + z 7 {\displaystyle x^{3}+3x^{2}y+z^{7}} is not homogeneous, because the sum of exponents does not match from term to term.
The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. [1] It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. The independent sets of ...