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]

^ Axler (2015) pp. 111-112, §§ 3.115, 3.119
^ ^a ^b Roman (2005) p. 48, § 1.16
^ Bourbaki, Algebra, ch. II, §10.12, p. 359
^ 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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[1] Axler (2015) pp. 111-112, §§ 3.115, 3.119

[:0-2] Roman (2005) p. 48, § 1.16

[3] Bourbaki, Algebra, ch. II, §10.12, p. 359

[mackiw-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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