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Concept
Row and column spaces
Summary
In linear algebra, the column space (also called the range or ) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the or range of the corresponding matrix transformation.
Let be a field. The column space of an m × n matrix with components from is a linear subspace of the m-space . The dimension of the column space is called the rank of the matrix and is at most min(m, n). A definition for matrices over a ring is also possible.
The row space is defined similarly.
The row space and the column space of a matrix A are sometimes denoted as C(AT) and C(A) respectively.
This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces and respectively.
Let A be an m-by-n matrix. Then
rank(A) = dim(rowsp(A)) = dim(colsp(A)),
rank(A) = number of pivots in any echelon form of A,
rank(A) = the maximum number of linearly independent rows or columns of A.
If one considers the matrix as a linear transformation from to , then the column space of the matrix equals the 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().
The concept of row space generalizes to matrices over , the field of complex numbers, or over 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 = Ax must reside in the column space of A. See singular value decomposition for more details on this second interpretation.
Given a matrix J:
the rows are
Consequently, the row space of J is the subspace of spanned by .
Since these four row vectors are linearly independent, the row space is 4-dimensional.
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