Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Direct sum of modules
Summary
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the notion.
The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces.
See the article decomposition of a module for a way to write a module as a direct sum of submodules.
We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by considering these two cases in depth.
Suppose V and W are vector spaces over the field K. The cartesian product V × W can be given the structure of a vector space over K by defining the operations componentwise:
(v1, w1) + (v2, w2) = (v1 + v2, w1 + w2)
α (v, w) = (α v, α w)
for v, v1, v2 ∈ V, w, w1, w2 ∈ 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 identified with V; similarly for {0} × W and W. (See internal direct sum below.) With this identification, every element of V ⊕ W can be written in one and only one way as the sum of an element of V and an element of W. The dimension of V ⊕ W is equal to the sum of the dimensions of V and W. One elementary use is the reconstruction
of a finite vector space from any subspace W and its orthogonal complement:
This construction readily generalizes to any finite number of vector spaces.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.