k×1

In linear algebra, a K×1 matrix denotes a matrix with K rows and 1 column. It is commonly interpreted as a column vector in a K-dimensional space. The parameter K is a positive integer that specifies the vector’s dimension. A typical notation uses entries x1, x2, ..., xK, written as a column vector x = [x1; x2; ...; xK] or x = (x1, x2, ..., xK)^T.

Operations and properties: If A is an m×K matrix and x is a K×1 vector, then the product A x is an m×1 vector. The inner product of two K×1 vectors x and y is x^T y, a scalar. Transposing a K×1 vector yields a 1×K row vector, denoted x^T. The ranking and linear independence of a set of K×1 vectors are central to many dimensionality and basis concepts in R^K.

Common uses: K×1 vectors represent coordinates of points in K-dimensional space, feature vectors in machine learning, and state or parameter vectors in dynamical systems and control theory. They are the standard form for column representations in linear systems, eigenvalue problems, and least squares analyses.

Special cases: When K = 1, a K×1 matrix reduces to a scalar value in a 1×1 container. For K > 1, a K×1 object is a genuine vector rather than a scalar and participates in matrix-vector products and vector algebra accordingly.

See also: vector, column vector, matrix, dimension.