Row and column spaces

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In mathematics, a matrix can be thought of as each row or column being a vector. Hence, a space formed by row 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 space of a matrix A is the set of all linear combinations of the columns in A. If A = [a₁, ...., a_n], then Col A = Span {a₁, ...., a_n}

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

Intuitively, given a matrix A, the action of the matrix A on a vector x will (1) first project x into the row space of A, (2) perform an invertible transformation, and (3) place the resulting vector y in the column space of A. Thus the result y =A x must reside in the column space of A.

[edit] 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 independent, 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.