In mathematics, 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. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Definition

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 v₁,...,v_n are vectors and a₁,...,a_n are scalars, then the linear combination of those vectors with those scalars as coefficients is A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... The concept of a scalar is used in mathematics and physics. ...

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 v₁,...,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). A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...

Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalisations below). 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 is no reason that n cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V. In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... 0 (zero), alternatively called naught, nil, ought, or nought, is both a number and a numeral. ...

Examples and counterexamples

Analytic geometry

Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R³. Consider the vectors e₁ := (1,0,0), e₂ := (0,1,0) and e₃ = (0,0,1). Then any vector in R³ is a linear combination of e₁, e₂ and e₃. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...

To see that this is so, take an arbitrary vector (a₁,a₂,a₃) in R³, and write:

Functional analysis

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: Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of negative one (−1), which cannot be represented by any real number. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Partial plot of a function f. ... In mathematics, the real line is simply the set of real numbers. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... The mathematical constant e is the base of the natural logarithm. ... In mathematics, the imaginary unit i (sometimes also represented by the Latin j or the Greek iota, but in this article i will be used exclusively) allows the real number system to be extended to the complex number system . ...

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On the other hand, the constant function 3 is not a linear combination of f and g. To see this, suppose that 3 could be written as a linear combination of e^it and e^−it. This means that there would exist complex scalars a and b such that ae^it + be^−it = 3 for all real numbers t. Setting t = 0 and t = π gives the equations a + b = 3 and a + b = −3, and clearly this cannot happen.

Algebraic geometry

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) p₁ := 1, p₂ := x + 1, and p₃ := x² + x + 1. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

Is the polynomial x² − 1 a linear combination of p₁, p₂, and p₃? To find out, consider an arbitrary linear combination of these vectors and try to see when it equals the desired vector x² − 1. Picking arbitrary coefficients a₁, a₂, and a₃, we want

Multiplying the polynomials out, this means

and collecting like powers of x, we get

Two polynomials are equal if and only if their corresponding coefficients are equal, so we can conclude ↔ ⇔ ≡ logical symbols representing iff. ...

This system of linear equations can easily be solved. First, the first equation simply says that a₃ is 1. Knowing that, we can solve the second equation for a₂, which comes out to −1. Finally, the last equation tells us that a₁ is also −1. Therefore, the only possible way to get a linear combination is with these coefficients. Indeed, In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ...

so x² − 1 is a linear combination of p₁, p₂, and p₃.

On the other hand, what about the polynomial x³ − 1? If we try to make this vector a linear combination of p₁, p₂, and p₃, then following the same process as before, we’ll get the equation

However, when we set corresponding coefficients equal in this case, the equation for x³ is

which is always false. Therefore, there is no way for this to work, and x³ − 1 is not a linear combination of p₁, p₂, and p₃.

The linear span

Main article: linear span In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ...

Take an arbitrary field K, an arbitrary vector space V, and let v₁,...,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, say S ={v₁,...,v_n}. We write the span of S as span(S) or sp(S): In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ...

Other related concepts

Sometimes, some single vector can be written in two different ways as a linear combination of v₁,...,v_n. If that is possible, then v₁,...,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. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

If S is linearly independent and the span of S equals V, then S is a basis for V. In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

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, cannot 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.

Generalisations

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 a₁v₁ + a₂v₂ + a₃v₃ + ..., going on forever. Such infinite linear combinations do not 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. In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard...

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. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ...

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 In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. ...

where a₁,...,a_n belong to K_L, b₁,...,b_n belong to K_R, and v₁,...,v_n belong to V.