Row and column spaces

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The row space and column space of an m-by-n matrix with real entries is the subspace of Rn generated by the row vectors and column vectors, respectiviely, of the matrix. Its dimension is equal to the rank of the matrix and is at most min(m,n).[1]

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Overview

Let A be a n by m matrix. Then

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 = [a1, ...., an], then colsp(A) = span {a1, ...., an}.

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.

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

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