Coordenasaddition

Coordenasaddition, also known as coordinate-wise addition or element-wise addition, is the operation of adding two objects by summing corresponding components. It is commonly defined for vectors in R^n or more generally for tuples in a product of algebraic structures.

Formally, given vectors x = (x1, ..., xn) and y = (y1, ..., yn) in R^n, the coordenasaddition is x ⊕ y = (x1 + y1, ..., xn + yn). The operation is well-defined whenever the underlying components can be added in their respective structures. It is commutative and associative, with the zero vector (0, ..., 0) as the identity element. Scalar multiplication distributes over coordenasaddition: a*(x ⊕ y) = (a x1, ..., a xn) ⊕ (a y1, ..., a yn).

Beyond vectors, coordenasaddition applies element-wise to matrices of the same size, and to arrays in programming languages, yielding a matrix or array of sums.

This operation is the standard addition in the Euclidean space R^n and is compatible with the vector space structure. It generalizes to any finite product of abelian groups, where addition is defined componentwise.

Examples: (3, -2, 5) ⊕ (1, 4, -3) = (4, 2, 2). In matrices: [[1,2],[3,4]] ⊕ [[5,6],[7,8]] = [[6,8],[10,12]].

Notes: In some contexts “coordinate-wise” is preferred or is referred to as element-wise or Hadamard addition. Caution is needed when coordinates do not form a vector space, as the operation may be ill-defined.