Row space

In linear algebra, the column space (also called the range or image) 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 image or range of the corresponding matrix transformation. In linear algebra, the column space (also called the range or image) 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 image or range of the corresponding matrix transformation. Let F {displaystyle mathbb {F} } be a field. The column space of an m × n matrix with components from F {displaystyle mathbb {F} } is a linear subspace of the m-space F m {displaystyle mathbb {F} ^{m}} . 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 K {displaystyle mathbb {K} } is also possible. The row space is defined similarly. This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces Rn and Rm respectively. Let A be an m-by-n 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 = , then colsp(A) = span {a1, ...., an}. The concept of row space generalizes to matrices over C, 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 = A x must reside in the column space of A. See singular value decomposition for more details on this second interpretation.