Editing Singular value decomposition (section)

===Range, null space and rank===
Another application of the SVD is that it provides an explicit representation of the [[Column space|range]] and [[null space]] of a matrix {{tmath|\mathbf M.}} The right-singular vectors corresponding to vanishing singular values of {{tmath|\mathbf M}} span the null space of {{tmath|\mathbf M}} and the left-singular vectors corresponding to the non-zero singular values of {{tmath|\mathbf M}} span the range of {{tmath|\mathbf M.}} For example, in the above [[#Example|example]] the null space is spanned by the last row of {{tmath|\mathbf V^*}} and the range is spanned by the first three columns of {{tmath|\mathbf U.}}

As a consequence, the [[rank of a matrix|rank]] of {{tmath|\mathbf M}} equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in <math>\mathbf \Sigma</math>. In numerical linear algebra the singular values can be used to determine the ''effective rank'' of a matrix, as [[rounding error]] may lead to small but non-zero singular values in a rank deficient matrix. Singular values beyond a significant gap are assumed to be numerically equivalent to zero.