Finiterank

Finite rank is a property of linear operators or matrices indicating that their images have finite dimension. If T is a linear map from a vector space V to a vector space W over a field, T has finite rank when the dimension of its image, Im(T), is finite. The rank of T, denoted rank(T), equals dim(Im(T)).

In matrix terms, a matrix A has finite rank equal to the dimension of its column space (or row space). The rank can be determined by row-reducing A and counting the number of nonzero rows, or by computing the number of linearly independent columns.

Common examples include projections onto a finite-dimensional subspace, inclusions from a finite-dimensional subspace, and matrices with only a finite number of linearly independent columns. Finite rank operators commonly arise in both finite- and infinite-dimensional contexts.

Key properties include:

- Relationship to nullity: For a linear map T between finite-dimensional spaces, dim(V) = rank(T) + nullity(T) (the rank-nullity theorem). In infinite-dimensional domains, rank can be finite even when nullity is infinite.

- Continuity and compactness: In normed spaces, every finite rank operator is bounded and hence continuous, and finite rank operators are compact.

- Decomposition: A finite-rank operator T can be expressed as a finite sum of rank-one operators, T = ∑_{i=1}^r x_i ⊗ f_i, with x_i in W and f_i in the dual space V*.

- Approximation: Finite rank operators often serve as simple approximations to more complex operators, and in many contexts they form a dense subset of the compact operators.

Finite rank concepts are central in linear algebra and functional analysis, providing a bridge between finite-dimensional intuition and infinite-dimensional applications.