Subvectors

A subvector is a vector obtained by selecting a subset of components from a given vector, while preserving their original order. If x is a vector in R^n and I is a subset of {1,...,n}, then the subvector x_I consists of the components x_i with i in I, listed in increasing order of i. If I consists of consecutive indices, the resulting vector is sometimes called a block or contiguous subvector.

Notation and construction: For x ∈ R^n and I ⊆ {1,...,n}, x_I ∈ R^|I| denotes the subvector determined by I. A subvector can be obtained by indexing (x_I = [x_i : i ∈ I]) or by applying a selection (projection) operator. If I = {i} then x_I is a scalar, one component of x.

Examples: Let x = (3, 5, 7, 2). Then x_{1,3} = (3, 7) and x_{2,4} = (5, 2). The subvectors x_{1} = 3 and x_{2,3} = (5, 7) illustrate single-component and multi-component cases.

In probability and statistics, subvectors are defined similarly for random vectors. If X is a random vector in R^n and I ⊆ {1,...,n}, the subvector X_I denotes the components with indices in I, and its distribution is the marginal distribution of those components.

Applications and notes: Subvectors are used in feature selection, dimensionality reduction, and block-structured analyses. They facilitate focus on specific coordinates or blocks of a vector, support modular reasoning about large vectors, and underpin projections in linear algebra and statistics.