vecN

vecN is a common notation in linear algebra for an N-dimensional vector. It refers to an element of the vector space F^N, where F is a field such as the real numbers. In many contexts vecN is written as v = (v1, v2, ..., vN) or as a column vector, and the index N conveys the number of components in the vector.

The standard representation uses a basis for vecN. The standard basis consists of N vectors e1, e2, ..., eN, where ei has a 1 in the i-th position and 0 elsewhere. Any vector v in vecN can be expressed as a linear combination v = sum over i of vi ei, where vi are the coordinates of v with respect to this basis. Different bases yield different coordinate representations for the same vector.

Operations on vecN include vector addition and scalar multiplication. The dot product (inner product) of u = (ui) and v = (vi) is <u, v> = sum over i of ui vi, which induces the Euclidean norm ||v|| = sqrt(<v, v>). Other norms and inner products can be defined, depending on context. The cross product is a specialized operation defined only in three dimensions.

Linear maps between vecN and vecM are represented by M×N matrices. Applying a matrix to a vecN vector yields a vecM vector, illustrating how linear transformations act on coordinate representations. Coordinate changes between bases are described by change-of-basis matrices.

vecN appears in diverse fields, including physics, computer graphics, and data science, where it is used to represent coordinates, feature vectors, or state variables. Example: in vec3, v = (1, -2, 3) has length sqrt(14) and can be dot-prod with w = (4, 0, -1) to yield 1.