Linear combination

Under a linear combination is understood in linear algebra, a vector which can be expressed by given vectors using the vector addition and scalar multiplication.

• 3.1 Positive coefficients
• 3.2 Affine combination
• 3.3 convex combination

Definition

Linear combinations of finitely many vectors

Be a vector space over the field. In addition, a finite number of vectors are given out. Then we call each vector in the form

Write leaves ( scalars ) is a linear combination of. The factors in the above table are called the coefficients of the linear combination. The presentation itself is referred to as a linear combination.

Example: In the three-dimensional ( real) vector space of vector is a linear combination of the vectors and, because

The coefficients are real numbers in this example, because here we consider a real vector space.

Linear combinations of any number of vectors

Linear combinations of infinite number of elements is considered only under the condition that in reality only a finite number thereof can be used in the sum.

Be a body and a vector space. It should also be indexed by the index set of the family of vectors. If you've finally to each a coefficient such that almost all coefficients are zero, then

The corresponding linear combination. That only a finite number of coefficients (and summands ) are different from 0, is required so that the sum may be defined in general. A convergent series is therefore not generally linear combination of its summands.

Linear combinations in left modules

In a further generalization of the notion of linear combination yields already makes sense if you look at rings instead of bodies and links modules instead of vector spaces. Many of the known from linear algebra, basic operations can also be performed in this generality, only solving for a vector from a linear combination may fail, because for this you have to multiply by the inverse of the coefficient in front of this vector and the ring contains these inverses in usually not.

General

In a vector space, the linear combination of vectors of coefficients from the body of the vector space again an element of the vector space. Let all the elements of the vector space as a linear combination of a set M represent, M is a generating system of the vector space. The set of all linear combinations of a set of vectors is called the linear hull.

The zero vector of a vector space can always be expressed as a linear combination of a given set of vectors. If all the coefficients of such a linear combination is equal to 0 ( zero element of the underlying body ), it is called a trivial linear combination. If the given vectors are linearly dependent, so can be written as non-trivial linear combination of the zero vector.

Linear combinations whose coefficients are arbitrary real or complex numbers, but integers (this is then also speaks of an integer linear combination ), play a central role in the extended Euclidean algorithm; it provides a representation of the greatest common divisor of two integers as a linear combination of and:

Special cases

Positive coefficients

The special linear combinations considered here, using an ordering on the coefficient body, so it is limited to: - or - vector spaces.

• If the coefficients of the linear combination of all greater than or equal to zero, then one speaks of a conic linear combination. The corresponding envelope is called conical shell and denoted by the symbol.
• If the coefficients of the linear combination of all strictly greater than zero, then one speaks of a positive combination.

Affine combination

• Is the sum of the coefficients equal to 1, it is a Affinkombination. This definition is possible for any links modules.

Convex combination

In real space is called a linear combination of convex if all the coefficients of the unit interval [0,1] are obtained and whose sum to 1:

This may account for the condition, because it is automatically determined by the sum of the non-negativity condition and the coefficients.

Convex combinations of convex combinations are convex combinations again. The set of all convex combinations of a given set of vectors is called the convex hull.