In linear algebra, the column space (also called the range or image) of a matrix A {\displaystyle A} is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation. Let F {\displaystyle F} be a field. The column space of an m × n {\displaystyle m\times n} matrix with components from F {\displaystyle F} is a linear subspace of the m-space F m {\displaystyle F^{m}} . The dimension of the column space is called the rank of the matrix and is at most min { m , n } {\displaystyle \min\{m,n\}} . A definition for matrices over a ring R {\displaystyle R} (i.e., with components from R {\displaystyle R} ) is also possible. The row space is defined similarly. The row space and the column space of a matrix A {\displaystyle A} are sometimes denoted as C ( A T ) {\displaystyle \operatorname {C} (A^{\mathsf {T}})} and C ( A ) {\displaystyle \operatorname {C} (A)} , respectively. This article considers matrices of real numbers. The row and column spaces are subspaces of the real coordinate spaces R n {\displaystyle \mathbb {R} ^{n}} and R m {\displaystyle \mathbb {R} ^{m}} , respectively.
Overview 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 the matrix represents a linear transformation, 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}). Given a matrix A, the action of the matrix A on a vector x returns a linear combination of the columns of A with the coordinates of x as coefficients; that is, the columns of the matrix generate the column space.
Example Given a matrix J:
J = [ 2 4 1 3 2 − 1 − 2 1 0 5 1 6 2 2 2 3 6 2 5 1 ] {\displaystyle J={\begin{bmatrix}2&4&1&3&2\\-1&-2&1&0&5\\1&6&2&2&2\\3&6&2&5&1\end{bmatrix}}}
the rows are
r 1 = [ 2 4 1 3 2 ] {\displaystyle \mathbf {r} _{1}={\begin{bmatrix}2&4&1&3&2\end{bmatrix}}} ,
r 2 = [ − 1 − 2 1 0 5 ] {\displaystyle \mathbf {r} _{2}={\begin{bmatrix}-1&-2&1&0&5\end{bmatrix}}} ,
r 3 = [ 1 6 2 2 2 ] {\displaystyle \mathbf {r} _{3}={\begin{bmatrix}1&6&2&2&2\end{bmatrix}}} ,
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