Row and Column Spaces - Overview

Overview

Let A be a n-by-m matrix. Then

1. rank(A) = dim(rowsp(A)) = dim(colsp(A)),

2. rank(A) = number of pivots in any echelon form of A,

3. rank(A) = the maximum number of linearly independent rows or columns of A.

If one considers the matrix as a linear transformation from Rn to Rm, 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 =, then colsp(A) = span {a₁, ...., a_n}.

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

Intuitively, given a matrix A, the action of the matrix A on a vector x will return a linear combination of the columns of A weighted by the coordinates of x as coefficients. Another way to look at this is that it 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. See the singular value decomposition for more details on this second interpretation.