Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics. As an example, consider the direct sum , where is the set of real numbers. is the Cartesian plane, the xy-plane from elementary algebra. In general, the direct sum of two objects is another object of the same type, so the direct sum of two geometric objects is a geometric object and the direct sum of two sets is a set.

To see how direct sum is used in abstract algebra, consider a more elementary structure in abstract algebra, the abelian group. The direct sum of two abelian groups and is another abelian group consisting of the ordered pairs where and . To add ordered pairs, we define the sum to be ; in other words addition is defined coordinate-wise. A similar process can be used to form the direct sum of any two algebraic structures, such as rings, modules, and vector spaces.

We can also form direct sums with any number of summands, for example , provided and are the same kinds of algebraic structures, that is, all groups or all rings or all vector spaces.

Matrix addition

In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered as a kind of addition for matrices, the direct sum and the Kronecker sum.

Entrywise sum

Two matrices must have an equal number of rows and columns to be added. The sum of two matrices A and B will be a matrix which has the same number of rows and columns as do A and B. The sum of A and B, denoted A + B, is computed by adding corresponding elements of A and B:

For example:

We can also subtract one matrix from another, as long as they have the same dimensions. A − B is computed by subtracting corresponding elements of A and B, and has the same dimensions as A and B. For example:

Direct sum

Another operation, which is used less often, is the direct sum (denoted by ⊕). Note the Kronecker sum is also denoted ⊕; the context should make the usage clear. The direct sum of any pair of matrices A of size m × n and B of size p × q is a matrix of size (m + p) × (n + q) defined as

Disjoint union (topology)

In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, two or more spaces may be considered together, each looking as it would alone.

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.

Definition

Let {X_i : i ∈ I} be a family of topological spaces indexed by I. Let

be the disjoint union of the underlying sets. For each i in I, let

be the canonical injection (defined by ). The disjoint union topology on X is defined as the finest topology on X for which the canonical injections {φ_i} are continuous.

Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage is open in X_i for each i ∈ I.