In mathematics, a matrix can be thought of being consisting of each row or column as a vector. Hence, a space formed by raw vectors or column vectors are said to be a row space or a column space.

The row space of an m-by-n matrix with real entries is the subspace of Rⁿ generated by the row vectors of the matrix. Its dimension is equal to the rank of the matrix and is at most min(m,n).

The column space of an m-by-n matrix with real entries is the subspace of R^m generated by the column vectors of the matrix. Its dimension is the rank of the matrix and is at most min(m,n).

If one considers the matrix as a linear transformation from Rⁿ to R^m, then the column space of the matrix equals the image of this linear transformation.

The column spaces of a matrix Z is the set of all linear combinations of the columns in Z. If Z = [a₁, ...., a_n], then Col Z = Span {a₁, ...., a_n}

The concept of row space generalises to matrices over any field, in particular C, the field of complex numbers.

Example

Given a matrix J:

the rows are r₁ = (2,4,1,3,2), r₂ = (−1,−2,1,0,5), r₃ = (1,6,2,2,2), r₄ = (3,6,2,5,1). Consequently the row space of J is the subspace of R⁵ spanned by { r₁, r₂, r₃, r₄ }. Since these four row vectors are linearly dependent, the row space is 4-dimensional. Moreover in this case it can be seen that they are all orthogonal to the vector n = (6,−1,4,−4,0), so it can be deduced that the row space consists of all vectors in R⁵ that are orthogonal to n.

See also null space.