Rank (linear algebra)

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]

  1. ^ Axler (2015) pp. 111-112, §§ 3.115, 3.119
  2. ^ a b Roman (2005) p. 48, § 1.16
  3. ^ Bourbaki, Algebra, ch. II, §10.12, p. 359
  4. ^ Mackiw, G. (1995), "A Note on the Equality of the Column and Row Rank of a Matrix", Mathematics Magazine, 68 (4): 285–286, doi:10.1080/0025570X.1995.11996337


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