How to rank linear functions

In linear algebra, the rank of a matrix

A

A is the dimension of the vector space generated (or spanned) by its columns.[1] This corresponds to the maximal number of linearly independent columns of

A

A. This, in turn, is identical to the dimension of the vector space spanned by its rows.[2] Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by

A

A. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics.

The rank is commonly denoted by

rank

⁡

(

A

)

{/displaystyle / operatorname {rank} (A) } or

rk

⁡

(

A

)

{/displaystyle / operatorname {rk} (A) }; sometimes the parentheses are not written, as in

rank

⁡

A

{/displaystyle / operatorname {rank} A}.