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The dimension of the column space is called the rank of the matrix and is at most min(m, n). [1] 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(A T) and C(A) respectively. [2] This article considers matrices of real numbers
A projective basis is + points in general position, in a projective space of dimension n. A convex basis of a polytope is the set of the vertices of its convex hull. A cone basis [5] consists of one point by edge of a polygonal cone. See also a Hilbert basis (linear programming).
Let the column rank of A be r, and let c 1, ..., c r be any basis for the column space of A. Place these as the columns of an m × r matrix C. Every column of A can be expressed as a linear combination of the r columns in C. This means that there is an r × n matrix R such that A = CR.
If is nonsingular, the columns indexed by B are a basis of the column space of . In this case, we call B a basis of the LP. Since the rank of A {\displaystyle A} is m , it has at least one basis; since A {\displaystyle A} has n columns, it has at most ( n m ) {\displaystyle {\binom {n}{m}}} bases.
The corresponding columns of the original matrix are a basis for the column space. See the article on column space for an example. This produces a basis for the column space that is a subset of the original column vectors. It works because the columns with pivots are a basis for the column space of the echelon form, and row reduction does not ...
A basis of the kernel of A consists in the non-zero columns of C such that the corresponding column of B is a zero column. In fact, the computation may be stopped as soon as the upper matrix is in column echelon form: the remainder of the computation consists in changing the basis of the vector space generated by the columns whose upper part is ...
where "old" and "new" refer respectively to the initially defined basis and the other basis, and are the column vectors of the coordinates of the same vector on the two bases. A {\displaystyle A} is the change-of-basis matrix (also called transition matrix ), which is the matrix whose columns are the coordinates of the new basis vectors on the ...
The solution can then be expressed as ^ = (), where is an matrix containing the first columns of the full orthonormal basis and where is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this x ^ {\displaystyle {\hat {\mathbf {x} }}} without explicitly inverting R 1 {\displaystyle R ...