High Quality Content by WIKIPEDIA articles! The column rank of a matrix A is the maximal number of linearly independent columns of A. Likewise, the row rank is the maximal number of linearly independent rows of A. Since the column rank and the row rank are always equal, they are simply called the rank of A. More abstractly, it is the dimension of the image of the linear transformation that is multiplication by A. For the proofs, see, e.g., Murase (1960), Andrea & Wong (1960), Williams & Cater (1968), Mackiw (1995). It is commonly denoted by either rk(A) or rank A. The rank of an m x n matrix is at most min(m, n). A matrix that has a rank as large as possible is said to have full rank; otherwise, the matrix is rank deficient. More generally, if a linear operator on a vector space (possibly infinite-dimensional) has finite-dimensional range (e.g., a finite rank operator), then the rank of the operator is defined as the dimension of the range.

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