User:Jim.belk/Draft:Reduced row echelon form

In linear algebra, a matrix is in reduced row echelon form (also known as reduced echelon form or row canonical form) if it satisfies the following requirements:* Any rows of zeros are at the bottom.* The first nonzero entry in any each is a one. (These entries are called pivots)* Each pivot lies to the right of the pivot in the preceding row.* All entries above a pivot an in the same column are zero.This is similar to the requirements for row echelon form, except that the entries above the pivots are required to be zero::$$\begin{pmatrix}1 & 0 & \ast & 0 & \ast & \ast & 0 & 0 & \ast \\0 & 1 & \ast & 0 & \ast & \ast & 0 & 0 & \ast \\0 & 0 &   0  & 1 & \ast & \ast & 0 & 0 & \ast \\0 & 0 &   0  & 0 &   0  &   0  & 1 & 0 & \ast \\0 & 0 &   0  & 0 &   0  &   0  & 0 & 1 & \ast \\0 & 0 &   0  & 0 &   0  &   0  & 0 & 0 &   0  \\0 & 0 &   0  & 0 &   0  &   0  & 0 & 0 &   0\end{pmatrix}$$Any matrix may be brought into reduced echelon form through row reduction, and the result does not depend on the specific sequence of elementary row operation used for the reduction. Stated differently, every matrix is row equivalent to a unique matrix in reduced echelon form.In linear algebra, reduced echelon form is the simplified form for a system of linear equations. It allows  as it allows for a natural parameterization of the solution set. For a homogeneous system, a basis for the nullspace of the matrix can be read off from the Reduced echelon form can also be used to express any linear subspace as the nullspace of a matrix, or to determine whether two linear subspaces are equal.==Systems of equations==When representing systems of linear equations using augmented matrices, reduced echelon form is the simplest form that a system may take. Each pivot column of the reduced row echelon form represents a dependent variable, while the columns without pivots are free variables.==See also==*Gaussian elimination*Gauss–Jordan elimination*Row echelon form== External links ==* MIT Linear Algebra Lecture on Row Reduced Form at Google Video, from MIT OpenCourseWare* Calculate the reduced row echelon form online*Module for Row Reduce Echelon Form Category:Numerical linear algebra