One also introduces the concept of the rank of a bilinear form (see Bilinear form) and the rank of a quadratic form (see Quadratic form). In the finite-dimensional case it coincides with the rank of a matrix of this mapping. The rank of a linear mapping is the dimension of the image under this mapping. The rank of a matrix does not change under multiplication by a non-singular matrix. The rank of a product of matrices is not greater than the rank of each of the factors. For a matrix over a field the rank is also equal to the maximal order of a non-zero minor. For matrices over a commutative ring with a unit these two concepts of rank coincide. 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). However, there exists another, unrelated, concept of rank in the theory of Lie algebras (see Rank of a Lie algebra). The rank of an algebra (over a skew-field) is understood to be the rank of its additive vector space. One defines the rank of an algebraic group and the rank of a Lie group in a special way. 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). Every commutative associative ring with a unit is such, so one can define, for example, the (Prüfer) rank of an Abelian group (which can be considered as a module over the ring $\mathbf Z$).
Rank of a matrix free#
If each free $R$-module has a unique rank, then $R$ is said to have the invariant basis number property. 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 Free module). For a module the situation is more complicated. 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). 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). Usually rank is defined either as the minimal 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. A concept closely connected with the concept of a basis.
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