# Rank (linear algebra)

## Dimension of the column space of a matrix / From Wikipedia, the free encyclopedia

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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*.^{[lower-roman 1]}