linguistic:paraphrasing, quality:correction, quality:corruption, stylistic:style_transfer
Computing ranks and bases

The Gaussian elimination algorithm can be applied to any m × n matrix A. In this way, for example, some 6 × 9 matrices can be transformed to a matrix that has a row echelon form like T =, {\displaystyle T={\begin{bmatrix}a&*&*&*&*&*&*&*&*\\0&0&b&*&*&*&*&*&*\\0&0&0&c&*&*&*&*&*\\0&0&0&0&0&0&d&*&*\\0&0&0&0&0&0&0&0&e\\0&0&0&0&0&0&0&0&0\end{bmatrix}},} where the stars are arbitrary entries, and a, b, c, d, e are nonzero entries. This echelon matrix T contains a wealth of information about A: the rank of A is 5, since there are 5 nonzero rows in T; the vector space spanned by the columns of A has a basis consisting of its columns 1, 3, 4, 7 and 9 (the columns with a, b, c, d, e in T), and the stars show how the other columns of A can be written as linear combinations of the basis columns.

All of this applies also to the reduced row echelon form, which is a particular row echelon format.