Linear combination

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

1 Definition
2 Examples and counterexamples
3 The linear span
4 Other related concepts
5 Generalizations

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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication 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

$a_1 v_1 + a_2 v_2 + a_3 v_3 + \cdots + a_n v_n \,$

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).

Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalisations below). In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. 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. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness 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.

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. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the real numbers may be described informally in several different ways 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.

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

$( a_1 , a_2 , a_3) = ( a_1 ,0,0) + (0, a_2 ,0) + (0,0, a_3) \,$

$= a_1 (1,0,0) + a_2 (0,1,0) + a_3 (0,0,1) \,$

$= a_1 e_1 + a_2 e_2 + a_3 e_3 \,$

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. For functional analysis as used in psychology see the Functional analysis (psychology article Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, the real line is simply the set R of singleton Real numbers However this term is usually used when R is to be treated as a In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis 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. The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line 71828. . . , and i is the imaginary unit, a square root of −1. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation ) Some linear combinations of f and g are:

.
$\cosh t = \begin{matrix}\frac12\end{matrix} e^{i t} + \begin{matrix}\frac12\end{matrix} e^{-i t} \,$
$2 \sin t = (-i ) e^{i t} + ( i ) e^{-i t} \,$

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. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Consider the vectors (polynomials) p₁ := 1, p₂ := x + 1, and p₃ := x² + x + 1.

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

$a_1 (1) + a_2 ( x + 1) + a_3 ( x^2 + x + 1) = x^2 - 1 \,$

Multiplying the polynomials out, this means

$( a_1 ) + ( a_2 x + a_2) + ( a_3 x^2 + a_3 x + a_3) = x^2 - 1 \,$

and collecting like powers of x, we get

$a_3 x^2 + ( a_2 + a_3 ) x + ( a_1 + a_2 + a_3 ) = 1 x^2 + 0 x + (-1) \,$

Two polynomials are equal if and only if their corresponding coefficients are equal, so we can conclude

$a_3 = 1, \quad a_2 + a_3 = 0, \quad a_1 + a_2 + a_3 = -1 \,$

This system of linear equations can easily be solved. ↔ In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example 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,

$x^2 - 1 = -1 - ( x + 1) + ( x^2 + x + 1) = - p_1 - p_2 + p_3 \,$

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

$0 x^3 + a_3 x^2 + ( a_2 + a_3 ) x + ( a_1 + a_2 + a_3 ) \,$

$= 1 x^3 + 0 x^2 + 0 x + (-1) \,$

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

$0 = 1 \,$

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

Take an arbitrary field K, an arbitrary vector space V, and let v₁,. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector . . ,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,. In the mathematical subfield of Linear algebra, the linear span, also called the linear hull, of a set of vectors in a Vector . . ,v_n}. We write the span of S as span(S) or sp(S):

$\mathrm{Sp}( v_1 ,\ldots, v_n) := \{ a_1 v_1 + \cdots + a_n v_n : a_1 ,\ldots, a_n \subseteq K \}. \,$

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. In Linear algebra, a family of vectors is linearly independent if none of them can be written as a Linear combination of finitely many other vectors 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. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

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.

Another related concept is the affine combination, which is a linear combination with the additional constraint that the coefficients a₁,. In Mathematics, an affine combination of vectors x 1. x n is vector \sum_{i=1}^{n}{\alpha_{i} \cdot . . ,a_n sum to unity.

Generalizations

If V is a topological vector space, then there may be a way to make sense of certain infinite linear combinations, using the topology of V. In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. 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.

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. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property The only difference is that we call spaces like V modules instead of vector spaces. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars 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 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 In that case, the most general linear combination looks like

$a_1 v_1 b_1 + \cdots + a_n v_n b_n \,$

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

Dictionary

-noun

(linear algebra) a sum, each of whose summands is an appropriate vector times an appropriate scalar (or ring element)

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