Linear combination

In linear algebra, a linear combination is understood to be a vector that can be expressed by given vectors using vector addition and scalar multiplication .

definition

Linear combinations of finitely many vectors

Let be a vector space over the body . In addition, a finite number are vectors from given. Then we call each vector that is in the form ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle v_{1},\ldots ,v_{n}}$ ${\displaystyle V}$ ${\displaystyle v\in V}$

${\displaystyle v=a_{1}v_{1}+a_{2}v_{2}+\dotsb +a_{n}v_{n}=\sum _{i=1}^{n}a_{i}v_{i}}$

can be written with scalars , a linear combination of . The factors in the above illustration are called the coefficients of the linear combination. The representation itself is also referred to as a linear combination. ${\displaystyle a_{1},\dotsc ,a_{n}\in K}$ ${\displaystyle v_{1},\dotsc ,v_{n}}$ ${\displaystyle a_{1},\dotsc ,a_{n}\in K}$

Example: In the three-dimensional ( real ) vector space the vector is a linear combination of the vectors and , because ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle v=\left(\!{\begin{smallmatrix}16\\-4\\3\end{smallmatrix}}\!\right)}$ ${\displaystyle v_{1}=\left(\!{\begin{smallmatrix}3\\-2\\4\end{smallmatrix}}\!\right)}$ ${\displaystyle v_{2}=\left(\!{\begin{smallmatrix}2\\0\\-1\end{smallmatrix}}\!\right)}$

${\displaystyle {\begin{pmatrix}16\\-4\\3\end{pmatrix}}=2{\begin{pmatrix}3\\-2\\4\end{pmatrix}}+5{\begin{pmatrix}2\\0\\-1\end{pmatrix}}.}$

The coefficients and in this example are real numbers, because is a real vector space. ${\displaystyle a_{1}=2}$ ${\displaystyle a_{2}=5}$ ${\displaystyle \mathbb {R} ^{3}}$

Linear combinations of any number of vectors

Linear combinations of infinitely many elements are only considered under the condition that in reality only a finite number of them are used in the sum.

Let be a body and a vector space. Furthermore, let us be a family of vectors indexed by the index set . If one then has a coefficient for each such that almost all coefficients are zero, then it is ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle (v_{i})_{i\in I}}$ ${\displaystyle I}$ ${\displaystyle v_{i}\in V}$ ${\displaystyle i\in I}$ ${\displaystyle a_{i}\in K}$

${\displaystyle v=\sum _{i\in I}a_{i}v_{i}}$

the associated linear combination. It is necessary that only finitely many coefficients (and thus summands) are different from 0 so that the sum can be defined at all. A convergent series is therefore generally not a linear combination of its summands.

Linear combinations in left modules

In a further generalization, the term linear combination already makes sense if one considers rings instead of bodies and left modules instead of vector spaces. Many of the simple operations known from linear algebra can also be carried out in this generality, only solving for a vector from a linear combination can fail, because for this one has to multiply by the inverse of the coefficient in front of this vector and the ring contains this inverse in usually not.

General

In a vector space , every linear combination of vectors is again an element of the vector space. The set of all linear combinations of a set of vectors is called its linear envelope ; it is always a subspace of . If all vectors in can be represented as a linear combination of a set , then there is a generating system of . ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle V}$

The zero vector of a vector space can always be expressed as a linear combination of a given set of vectors. If all coefficients of such a linear combination are equal to 0 (zero element of the underlying body), one speaks of a trivial linear combination. If the given vectors are linearly dependent , the zero vector can also be written as a non-trivial linear combination. In general, the coefficients of a linear combination of vectors are uniquely determined precisely when the vectors are linearly independent.

Linear combinations, the coefficients of which are not arbitrary real or complex numbers, but whole numbers (one also speaks of an integral linear combination ), play a central role in the extended Euclidean algorithm ; it provides a representation of the greatest common divisor of two whole numbers as a linear combination of and : ${\displaystyle a,b}$ ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \operatorname {ggT} (a,b)=s\cdot a+t\cdot b}$.

Special cases

The special linear combinations considered here use an order on the coefficient field, they are therefore limited to - or - vector spaces. ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$

Positive coefficients

→ Main article : Conical combination

• If the coefficients of the linear combination are all greater than or equal to zero, one speaks of a conical linear combination . If the coefficients of the linear combination are all really greater than zero, one speaks of a positive combination . ${\displaystyle a_{i}}$

Affine combination

• If the sum of the coefficients is 1, it is an affine combination . This definition is possible for any link module.

Convex combination

In real spaces a linear combination is called a convex combination if all the coefficients come from the unit interval [0,1] and their sum is 1:

${\displaystyle v=a_{1}v_{1}+a_{2}v_{2}+\dotsb +a_{n}v_{n}=\sum _{i=1}^{n}a_{i}v_{i},\quad 0\leq a_{i}\leq 1,\quad \sum _{i=1}^{n}a_{i}=1}$.

The condition can be omitted, because it results automatically from the cumulative condition and the non-negativity of the coefficients. With the above notation, the following applies in real spaces: A linear combination is a convex combination if and only if it is conical and affine. ${\displaystyle a_{i}\leq 1}$

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

literature

• Gerd Fischer : Linear Algebra. 14th, revised edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03217-0 .