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Linear algebra

Linear algebra is sub-discipline of mathematics concerning vector spaces often countably infinite or finite dimensional, with a linear mappings between such spaces. Such an investigation is aggravated initially by a system of linear equations in various unknowns. Such equations are represented logically using the formalism of matrices and vectors. This area is central to both applied and pure mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, which leads to a number of generalizations. Functional analysis is concerned with the infinite dimensional version of the theory of vector spaces. As Combined with calculus, linear algebra facilitates the solution of linear systems of Linear algebras. Methods from linear algebra are applied in analytic physics, geometry, natural sciences, engineering, the social sciences principally in economics and computer science. Because of linear algebra is such a well-developed theory, nonlinear mathematical models are infrequently approximated by linear ones.

 

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Vector spaces:

The important parts of linear algebra are vector spaces. A vector space over a field Q is a set V together with two binary operations. Elements of V are known elements and vectors of Q are called scalars. The 1st operation, "vector addition" takes any two vectors v and w and outputs a third vector v + w. The second operation takes any scalar a and any vector v and outputs a new vector av. In the first case, where the multiplication is done by rescaling the vector v by a scalar a, the multiplication is known as scalar multiplication of v by a.

The operations of addition and multiplication in a vector space represents in the following axioms. In this list let u, v and w be arbitrary vectors in V, and a  and b scalars in Q.

Axiom

Signification

Associativity of addition

u + (v + w) = (u + v) + w

Commutativity of addition

u + v = v + u

Identity element of addition

There exists an element 0 ∈ V, called the 

zero vector, such that v + 0 = v for all v ∈ V.

Inverse elements of addition

For every v ∈ V, there exists an element -v ∈ V, called the

 additive inverse of v, such that v + (-v) = 0

Distributivity of scalar multiplication with respect to vector addition

a(u + v) = au + av

Distributivity of scalar multiplication with respect to field addition

(a + b)v = av + bv

Compatibility of scalar multiplication with field multiplication

a(bv) = (ab)v [nb 1]

Identity element of scalar multiplication

1v = v, where 1 denotes the multiplicative identity in Q.

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