View source for Rank

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A concept closely connected with the concept of a [[Basis|basis]]. Usually rank is defined either as the minimal [[Cardinality|cardinality]] of a generating set (in this way, for example, one introduces the basis rank of an algebraic system), or as the maximal cardinality of a subsystem of elements which are independent in a certain sense.

The rank of a system of a vectors in a vector space over a skew-field is the maximal number of linearly independent vectors in this system (see [[Linear independence|Linear independence]]). The rank, or dimension, of a vector space, in particular, is equal to the number of elements in a basis of this space (the rank does not depend on the choice of the basis: all bases have the same cardinality). For a module the situation is more complicated. There exist associative rings $R$ such that even a free module over $R$ can have two bases with a different number of elements (see [[Rank of a module|Rank of a module]]; [[Free module|Free module]]). If each free $R$-module has a unique rank, then $R$ is said to have the invariant basis number property. Every commutative associative ring with a unit is such, so one can define, for example, the (Prüfer) rank of an [[Abelian group|Abelian group]] (which can be considered as a module over the ring $\mathbf Z$). In the non-Abelian case two concepts of the rank of a group are introduced, the general and the special rank (see [[Rank of a group|Rank of a group]]). One defines the [[Rank of an algebraic group|rank of an algebraic group]] and the [[Rank of a Lie group|rank of a Lie group]] in a special way.

The rank of an algebra (over a skew-field) is understood to be the rank of its additive vector space. However, there exists another, unrelated, concept of rank in the theory of Lie algebras (see [[Rank of a Lie algebra|Rank of a Lie algebra]]).

The rank of a matrix is defined as the rank of the system of vectors forming its rows (row rank) or of the system of columns (column rank). For matrices over a commutative ring with a unit these two concepts of rank coincide. For a matrix over a field the rank is also equal to the maximal order of a non-zero [[Minor|minor]]. The rank of a product of matrices is not greater than the rank of each of the factors. The rank of a matrix does not change under multiplication by a non-singular matrix.

The rank of a linear mapping is the dimension of the image under this mapping. In the finite-dimensional case it coincides with the rank of a matrix of this mapping.

One also introduces the concept of the rank of a bilinear form (see [[Bilinear form|Bilinear form]]) and the rank of a quadratic form (see [[Quadratic form|Quadratic form]]). They also (in the finite-dimensional case) coincide with the rank of a matrix of the corresponding form.



====Comments====
The invariant basis number property is also called the invariant dimension property.

For matrices over skew-fields one considers two ranks, which need not be equal: 1) the rank of the rows considered in a left vector space (it equals the rank of the columns considered in a right vector space); and 2) the rank obtained by interchanging "left" and "right" in 1).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.M. Cohn, "Algebra" , '''1–3''' , Wiley (1988) {{MR|0949350}} {{ZBL|1079.16013}} {{ZBL|1006.00001}} {{ZBL|1003.00001}} {{ZBL|1031.16018}} {{ZBL|1016.20033}} {{ZBL|0952.00001}} {{ZBL|0809.15001}} {{ZBL|0719.00002}} {{ZBL|0674.10023}} {{ZBL|0703.00002}} {{ZBL|0481.00001}} {{ZBL|0461.08001}} {{ZBL|0341.00002}} {{ZBL|0314.68032}} {{ZBL|0272.00003}} {{ZBL|0217.50302}} {{ZBL|0169.32601}} {{ZBL|0141.01002}} {{ZBL|0127.26402}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra I" , Addison-Wesley (1974) pp. Chapt. II (Translated from French) {{MR|0354207}} {{ZBL|0281.00006}} </TD></TR></table>