- Row and column spaces
The row space of an "m"-by-"n" matrix with real entries is the subspace of R"n" 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 R"n" 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, ...., an] , then Col A = Span {a1, ...., an}
The concept of row space generalises to matrices over any field, in particular C, the field of
complex number s.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.
Example
Given a matrix J::the rows arer1 = (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 .External links
* [http://mfile.akamai.com/7870/rm/mitstorage.download.akamai.com/7870/18/18.06/videolectures/strang-1806-lec06-20sep1999-80k.rm Lecture on column space and nullspace by Gilbert Strang of MIT]
Wikimedia Foundation. 2010.
