In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns.[1][2][3] This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows.[4] Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by 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) or rk(A);[2] sometimes the parentheses are not written, as in rank A.[i]
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