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This article includes a list of references, but its sources remain unclear because it has insufficient inline citations . Please help to improve this article by introducing more precise citations. ( May 2015 ) (Learn how and when to remove this template message) Algebraic structures Group-like Group Semigroup / Monoid Rack and quandle Quasigroup and loop Abelian group Magma Lie group Group theory Ring-like Ring Semiring Near-ring Commutative ring Integral domain Field Division ring Ring theory Lattice-like Lattice Semilattice Complemented lattice Total order Heyting algebra Boolean algebra Map of lattices Lattice theory Module-like Module Group with operators Vector space Linear algebra Algebra-like Algebra Associative Non-associative Composition algebra Lie algebra Graded Bialgebra v t e In mathematics, a module is one of the fundamental algebraic stru...

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field. [1] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, [a] and all bases of a vector space have equal cardinality; [b] as a result, the dimension of a vector space is uniquely defined. We say V is finite-dimensional if the dimension of V is finite, and infinite-dimensional if its dimension is infinite. The dimension of the vector space V over the field F can be written as dim F ( V ) or as [V : F], read "dimension of V over F ". When F can be inferred from context, dim( V ) is typically written. Contents 1 Examples 2 Facts 3 Generalizations 3.1 Trace 4 See also 5 Notes 6 References 7 External links Examples The vector space R 3 has {(100),(010),(00...

This article is about the characteristic polynomial of a matrix or of an endomorphism of vector spaces. For the characteristic polynomial of a matroid, see Matroid. For that of a graded poset, see Graded poset. In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix as coefficients. The characteristic polynomial of an endomorphism of vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial. The characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix. It is a graph invariant, though it is not complete: the smallest pair of non-isomorphic graphs with the same characteristic polynomial...

For the minimal polynomial of an algebraic element of a field, see Minimal polynomial (field theory). In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P ( A ) = 0 . Any other polynomial Q with Q ( A ) = 0 is a (polynomial) multiple of μ A . The following three statements are equivalent: λ is a root of μ A , λ is a root of the characteristic polynomial χ A of A , λ is an eigenvalue of matrix A . The multiplicity of a root λ of μ A is the largest power m such that Ker(( A − λI n ) m ) strictly contains Ker(( A − λI n ) m −1 ) . In other words, increasing the exponent up to m will give ever larger kernels, but further increasing the exponent beyond m will just give the same kernel. If the field F is not algebraically closed, then the minimal and characteristic polynomials need not factor according to their roots (in F ) alone, in other words they may have ...