Editing Rank (linear algebra) (section)

=== Decomposition rank ===
The rank of {{mvar|A}} is the smallest positive integer {{mvar|k}} such that {{mvar|A}} can be factored as <math>A = CR</math>, where {{mvar|C}} is an {{math|''m'' × ''k''}} matrix and {{mvar|R}} is a {{math|''k'' × ''n''}} matrix. In fact, for all integers {{mvar|k}}, the following are equivalent:

# the column rank of {{mvar|A}} is less than or equal to {{mvar|k}},
# there exist {{mvar|k}} columns <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math> of size {{mvar|m}} such that every column of {{mvar|A}} is a linear combination of <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math>,
# there exist an <math>m \times k</math> matrix {{mvar|C}} and a <math>k \times n</math> matrix {{mvar|R}} such that <math>A = CR</math> (when {{mvar|k}} is the rank, this is a [[rank factorization]] of {{mvar|A}}),
# there exist {{mvar|k}} rows <math>\mathbf{r}_1,\ldots,\mathbf{r}_k</math> of size {{mvar|n}} such that every row of {{mvar|A}} is a linear combination of <math>\mathbf{r}_1,\ldots,\mathbf{r}_k</math>,
# the row rank of {{mvar|A}} is less than or equal to {{mvar|k}}.

Indeed, the following equivalences are obvious: <math>(1)\Leftrightarrow(2)\Leftrightarrow(3)\Leftrightarrow(4)\Leftrightarrow(5)</math>.
For example, to prove (3) from (2), take {{mvar|C}} to be the matrix whose columns are <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math> from (2).
To prove (2) from (3), take <math>\mathbf{c}_1,\ldots,\mathbf{c}_k</math> to be the columns of {{mvar|C}}.

It follows from the equivalence <math>(1)\Leftrightarrow(5)</math> that the row rank is equal to the column rank.

As in the case of the "dimension of image" characterization, this can be generalized to a definition of the rank of any linear map: the rank of a linear map {{math|''f'' : ''V'' → ''W''}} is the minimal dimension {{mvar|k}} of an intermediate space {{mvar|X}} such that {{mvar|f}} can be written as the composition of a map {{math|''V'' → ''X''}} and a map {{math|''X'' → ''W''}}. Unfortunately, this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions). See [[rank factorization]] for details.