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Linear algebra is a branch of
that deals with vector spaces, also known linear spaces, along with linear functions in which input one vector or output another. These functions are called linear maps or linear operators or linear transformations and these can be represented by matrices if a basis is given. The matrix theory is considered in a part of linear algebra. Linear algebra is generally restricted to the case of finite vector spaces, dimensional, while the peculiarities of the infinite dimensional case are covered in linear functional analysis.
The subject linear algebra is central to modern mathematics or its applications. An elementary application of linear algebra is to find the answers of a system of linear equations in several unknowns. Many other advanced applications are ubiquitous in areas as diverse as functional analysis and abstract algebra. Linear algebra has a concrete representation in analytic geometry and is generalized in module theory and in operator theory. It has extensive applications in engineering, natural sciences, physics, computer science, and the social sciences. Nonlinear mathematical models may often be approximated by linear ones.
The main structures of linear algebra are linear maps and vector spaces between them. A vector space is a set that elements may be added together and multiplied by the scalars, or numbers. In several physical applications, the scalars are known real numbers, R. More commonly, the scalars may form any field F—in short one can consider vector spaces over the field Q of rational numbers, the field C of complex numbers, or a finite field Fq. These two operations must behave similarly to the addition and multiplication of numbers: addition is associative and commutative and multiplication distributes over addition, and so on, the two operations must satisfy a list of axioms picked to emulate the property of addition and scalar multiplication of Euclidean vectors in the coordinate n-space R
. One of the axioms stipulates the existence of zero vector, that behaves analogously to the number zero with respect addition. Elements of a general vector space V can be objects of any nature, for i.e., polynomials and functions , but when viewed as elements of V, these are frequently known vectors.
The two vector spaces V and W over a field F given, a linear transformation is a map
T: V -> W
Which is compatible with addition and scalar multiplication:
T(u+v) = T(u)+ T(v)
T(rv) = rT(v)
For any vectors u,v (- V and a scalar r (- F.
A fundamental role in linear algebra is played by the notions of linear combination, linear independence and span of vectors and basis and the dimension of a vector space. Given a vector space V over a field F, an expression of the form
, …, v
, …, r
are scalars, is known the linear combination of the vectors v
, …, v
with coefficients r
, …, r
The set of all linear combinations of vectors v
, …, v
is known their span.
Main useful theorem
- (AC) Every vector space has a basis dimension of a vector space is well-defined.
- A matrix is known invertible or non-singular, if and only if given linear map represented by the matrix is an isomorphism.
- Any vector space over a field F of dimension n is isomorphic to Fn as a vector space over F.
- Corollary: Any two vector spaces over F of the same finite dimension are known isomorphic to each other.
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