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An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range (or image) and the set of elements that are mapped to the zero vector, called the kernel of the map.
In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation .
In linear algebra, two n-by-n matrices A and B are called similar if there exists an invertible n-by-n matrix P such that =. Similar matrices represent the same linear map under two (possibly) different bases, with P being the change-of-basis matrix.
This glossary of linear algebra is a list of definitions and terms relevant to the field of linear algebra, the branch of mathematics concerned with linear equations and their representations as vector spaces. For a glossary related to the generalization of vector spaces through modules, see glossary of module theory
Then a linear algebraic group G over an algebraically closed field k is a subgroup G(k) of the abstract group GL(n,k) for some natural number n such that G(k) is defined by the vanishing of some set of regular functions. For an arbitrary field k, algebraic varieties over k are defined as a special case of schemes over k.
The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.
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Kernel and image of a linear map L from V to W. The kernel of L is a linear subspace of the domain V. [3] [2] In the linear map :, two elements of V have the same image in W if and only if their difference lies in the kernel of L, that is, = () =.