Linear combinations are a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalisations given at the end of the article.

Suppose that K is a field and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars. If v_{1},...,v_{n} are vectors and a_{1},...,a_{n} are scalars, then the linear combination of those vectors with those scalars as coefficients is
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_{1},...,v_{n}, 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).
Note that you can only take a linear combination of finitely many vectors (except as described in Generalisations below); that is, the number n may be finite. However, the set S that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors. Also, there's no reason that the finite number n can't be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V.
Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R^{3}. Consider the vectors e_{1} := (1,0,0), e_{2} := (0,1,0) and e_{3} = (0,0,1). Then any vector in R^{3} is a linear combination of e_{1}, e_{2} and e_{3}.
To see that this is so, take an arbitrary vector (a_{1},a_{2},a_{3}) in R^{3}, and write:
Let K be the set C of all complex numbers, and let V be the set C_{C}(R) of all continuous functions from the real line R to the complex plane C. Consider the vectors (functions) f and g defined by f(t) := e^{it} and g(t) := e^{−it}. (Here, e is the base of the natural logarithm, about 2.71828..., and i is the imaginary unit, a square root of −1.) Some linear combinations of f and g are:
Let K be any field (R, C, or whatever you like best), and let V be the set P of all polynomials with coefficients taken from the field K. Consider the vectors (polynomials) p_{1} := 1, p_{2} := x + 1, and p_{3} := x^{2} + x + 1.
Is the polynomial x^{2} − 1 a linear combination of p_{1}, p_{2}, and p_{3}? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x^{2} − 1. Picking arbitrary coefficients a_{1}, a_{2}, and a_{3}, we want
On the other hand, what about the polynomial x^{3} − 1? If we try to make this vector a linear combination of p_{1}, p_{2}, and p_{3}, then following the same process as before, we'll get the equation
Take an arbitrary field K, an arbitrary vector space V, and let v_{1},...,v_{n} be vectors (in V). It's interesting to consider the set of all linear combinations of these vectors. This set is called the linear span (or just span) of the vectors v_{1},...,v_{n}. In symbols,
Theorem 1: Sp(v_{1},...,v_{n}) is a subspace of V. Furthermore, this span is the smallest subspace of V that the vectors v_{1},...,v_{n} all belong to.
This fact (which is proved later in this section) is one reason why the span is important.
Now let S be a subset of the vector space V. The linear span of S consists of all linear combinations of elements of S. In symbols,
Theorem 2: Sp(S) is also a subspace of V. Furthermore, this span is the smallest subspace of V that is a superset of S.
The rest of this section is a proof of Theorem 1. Theorem 2 is very similar, but it's a bit messier to write down, since the vectors involved in any given linear combination can vary.
Proof of Theorem 1:
Property 1:
The most general possible two elements of the span are x := a_{1}v_{1} + ... + a_{n}v_{n} and y := b_{1}v_{1} + ... + b_{n}v_{n}.
We have to show that x + y is also a linear combination.
By using associativity and commutativity of addition and the distributive law, we can write
Property 2:
Let c be a scalar and again take x := a_{1}v_{1} + ... + a_{n}v_{n}.
We have to show that cx is also a linear combination.
Now,
Property 3:
The zero element 0_{V} of V is a linear combination because we can write
Minimality:
Suppose W is another subspace of V which contains the vectors v_{1},...,v_{n}.
Then W is closed under scalar multiplication and addition of vectors, so we can prove by mathematical induction that a_{1}v_{1} + ... + a_{n}v_{n} is an element of W for any scalars a_{1},...,a_{n}.
Thus, sp(v_{1},...,v_{n}), the set of all such linear combinations, is a subset of W.
Sometimes, some single vector can be written in two different ways as a linear combination of v_{1},...,v_{n}. If that is possible, then v_{1},...,v_{n} are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors.
If S is linearly independent and the span of S equals V, then S is a basis for V.
We can think of linear combinations as the most general sort of operation on a vector space. The basic operations of addition and scalar multiplication, together with the existence of an additive identity and additive inverses, can't be combined in any more complicated way than the generic linear combination. Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces.
If V is a topological vector space, then there may be a way to make sense of certain infinite linear combination, using the topology of V. For example, we might be able to speak of a_{1}v_{1} + a_{2}v_{2} + a_{3}v_{3} + ..., going on forever. Such infinite linear combinations don't always make sense; we call them convergent when they do. Allowing more linear combinations in this case can also lead to a different concept of span, linear independence, and basis. The articles on the various flavours of topological vector spaces go into more detail about these.
If K is a commutative ring instead of a field, then everything that has been said above about linear combinations generalises to this case without change. The only difference is that we call spaces like V modules instead of vector spaces. If K is a noncommutative ring, then the concept still generalises, with one caveat: Since modules over noncommutative rings come in left and right versions, our linear combinations may also come in either of these versions, whatever is appropriate for the given module. This is simply a matter of doing scalar multiplication on the correct side.
A more complicated twist comes when V is a bimodule[?] over two rings, K_{L} and K_{R}. In that case, the most general linear combination looks like
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