VectorN

VectorN is a standard term used to refer to a vector with N components in a vector space. Formally, a vector in N dimensions is written as v = (v1, v2, ..., vN) and is an element of F^N, where F is a field (most commonly the real numbers R).

Operations on VectorN are defined componentwise: vector addition v + w equals (v1 + w1, ..., vN + wN), and scalar multiplication a v equals (a v1, ..., a vN) for any a in F. The standard inner product on R^N is ⟨v, w⟩ = ∑ vi wi, with the associated Euclidean norm ||v|| = sqrt(⟨v, v⟩). These structures make VectorN an inner product space and, when F = R, a Euclidean space.

Basis and dimension are central in the study of VectorN. The space R^N has the standard basis e1, e2, ..., eN, where ei has a 1 in the i-th position and 0 elsewhere. Any VectorN can be expressed as a linear combination of basis vectors. The dimension of the space is N, and vectors are linearly independent if the only solution to a1 v1 + ... + aN vN = 0 is a1 = ... = aN = 0.

Applications of VectorN span many fields. As a compact representation for coordinates, VectorN underpins high-dimensional data analysis and machine learning where a feature vector of length N is modeled as a VectorN. In physics and computer graphics, vectors describe quantities with magnitude and direction in N-dimensional spaces, enabling operations such as projection, rotation, and transformation.

Extensions of VectorN include using different norms (Lp norms), complex vectors in C^N, and generalized coordinate systems. The concept also connects to broader ideas in linear algebra, such as bases, subspaces, and linear transformations. See also: Euclidean space, norm, inner product, basis, coordinate system.