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For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups is ...
The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory : the direct sum is the coproduct , while the direct product is the product.
The direct sum is a submodule of the direct product of the modules M i (Bourbaki 1989, §II.1.7). The direct product is the set of all functions α from I to the disjoint union of the modules M i with α(i)∈M i, but not necessarily vanishing for all but finitely many i. If the index set I is finite, then the direct sum and the direct product ...
In the natural sciences, a vector quantity (also known as a vector physical quantity, physical vector, or simply vector) is a vector-valued physical quantity. [9] [10] It is typically formulated as the product of a unit of measurement and a vector numerical value (), often a Euclidean vector with magnitude and direction.
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
For any non-negative integer n, =, the cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number, then its rank is n. A direct sum of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group).
The inner product of two vectors (also known as the scalar product, not to be confused with scalar multiplication) is represented as an ordered pair enclosed in angle brackets. The inner product of two vectors u and v would be represented as: u , v {\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle }