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The xy-plane, a two-dimensional vector space, can be thought of as the direct sum of two one-dimensional vector spaces, namely the x and y axes. In this direct sum, the x and y axes intersect only at the origin (the zero vector).
In the Cartesian plane, these pairs lie on a hyperbola, and when the double sum is fully expanded, there is a bijection between the terms of the sum and the lattice points in the first quadrant on the hyperbolas of the form xy = k, where k runs over the integers 1 ≤ k ≤ n: for each such point (x,y), the sum contains a term g(x)h(y), and ...
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 ...
The vector space is said to be the algebraic direct sum (or direct sum in the category of vector spaces) when any of the following equivalent conditions are satisfied: The addition map S : M × N → X {\\displaystyle S:M\\times N\\to X} is a vector space isomorphism .
(x 1, y 1) + (x 2, y 2) = (x 1 + x 2, y 1 + y 2). Let R + be the group of positive real numbers under multiplication. Then the direct product R + × R + is the group of all vectors in the first quadrant under the operation of component-wise multiplication (x 1, y 1) × (x 2, y 2) = (x 1 × x 2, y 1 × y 2). Let G and H be cyclic groups with two ...
In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices. In general, the direct sum of n ...
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 dual notion.
The Jordan identity implies that if x and y are elements of A, then the endomorphism sending z to x(yz)−y(xz) is a derivation. Thus the direct sum of A and der(A) can be made into a Lie algebra, called the structure algebra of A, str(A). A simple example is provided by the Hermitian Jordan algebras H(A,σ).