Editing Linear combination (section)

== Definition==
Let ''V'' be a [[vector space]] over the field ''K''. As usual, we call elements of ''V'' ''[[vector space|vector]]s'' and call elements of ''K'' ''[[scalar (mathematics)|scalars]]''.
If '''v'''<sub>1</sub>,...,'''v'''<sub>''n''</sub> are vectors and ''a''<sub>1</sub>,...,''a''<sub>''n''</sub> are scalars, then the ''linear combination of those vectors with those scalars as coefficients'' is
:<math>a_1 \mathbf v_1 + a_2 \mathbf v_2 + a_3 \mathbf v_3 + \cdots + a_n \mathbf v_n.</math>
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of '''v'''<sub>1</sub>,...,'''v'''<sub>''n''</sub> always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of [[linear dependence]]: a family ''F'' of vectors is linearly independent precisely if any linear combination of the vectors in ''F'' (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each '''v'''<sub>''i''</sub>; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.

In a given situation, ''K'' and ''V'' may be specified explicitly, or they may be obvious from context. In that case, we often speak of ''a linear combination of the vectors'' '''v'''<sub>1</sub>,...,'''v'''<sub>''n''</sub>, with the coefficients unspecified (except that they must belong to ''K''). Or, if ''S'' is a [[subset]] of ''V'', we may speak of ''a linear combination of vectors in S'', where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set ''S'' (and the coefficients must belong to ''K''). Finally, we may speak simply of ''a linear combination'', where nothing is specified (except that the vectors must belong to ''V'' and the coefficients must belong to ''K''); in this case one is probably referring to the expression, since every vector in ''V'' is certainly the value of some linear combination.

Note that by definition, a linear combination involves only [[finite set|finite]]ly many vectors (except as described in the {{section link||Generalizations}} section.
However, the set ''S'' that the vectors are taken from (if one is mentioned) can still be [[Infinity|infinite]]; each individual linear combination will only involve finitely many vectors.
Also, there is no reason that ''n'' cannot be [[0 (number)|zero]]; in that case, we declare by convention that the result of the linear combination is the [[zero vector]] in ''V''.