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Linear combination

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Sum of terms, each multiplied with a scalar

"Superposition" redirects here. For other uses, seesuperposition (disambiguation).

$\mathbf {v}$ is the linear combination of vectors $\mathbf {u} _{1}$ and $\mathbf {u} _{2}$ such that $\mathbf {v} =2\cdot \mathbf {u} _{1}+1.5\cdot \mathbf {u} _{2}$

{\displaystyle \mathbf {v} } — $\mathbf {v}$ is the linear combination of vectors $\mathbf {u} _{1}$ and $\mathbf {u} _{2}$ such that $\mathbf {v} =2\cdot \mathbf {u} _{1}+1.5\cdot \mathbf {u} _{2}$

Inmathematics, alinear combination orsuperposition is anexpression constructed from aset of terms by multiplying each term by a constant and adding the results (e.g. a linear combination ofx andy would be any expression of the formax +by, wherea andb are constants).^[1]^[2]^[3] The concept of linear combinations is central tolinear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of avector space over afield, with some generalizations given at the end of the article.

Definition

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LetV be avector space over the fieldK. As usual, we call elements ofVvectors and call elements ofKscalars.Ifv₁,...,v_n are vectors anda₁,...,a_n are scalars, then thelinear combination of those vectors with those scalars as coefficients is

a_{1}\mathbf {v} _{1}+a_{2}\mathbf {v} _{2}+a_{3}\mathbf {v} _{3}+\cdots +a_{n}\mathbf {v} _{n}.

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 ofv₁,...,v_n 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 oflinear dependence: a familyF of vectors is linearly independent precisely if any linear combination of the vectors inF (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 eachv_i; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.

In a given situation,K andV may be specified explicitly, or they may be obvious from context. In that case, we often speak ofa linear combination of the vectorsv₁,...,v_n, with the coefficients unspecified (except that they must belong toK). Or, ifS is asubset ofV, we may speak ofa linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the setS (and the coefficients must belong toK). Finally, we may speak simply ofa linear combination, where nothing is specified (except that the vectors must belong toV and the coefficients must belong toK); in this case one is probably referring to the expression, since every vector inV is certainly the value of some linear combination.

Note that by definition, a linear combination involves onlyfinitely many vectors (except as described in the§ Generalizations section.However, the setS that the vectors are taken from (if one is mentioned) can still beinfinite; each individual linear combination will only involve finitely many vectors.Also, there is no reason thatn cannot bezero; in that case, we declare by convention that the result of the linear combination is thezero vector inV.

Examples and counterexamples

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Euclidean vectors

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Let the fieldK be the setR ofreal numbers, and let the vector spaceV be theEuclidean spaceR³.Consider the vectorse₁ = (1,0,0),e₂ = (0,1,0) ande₃ = (0,0,1).Thenanyvector inR³ is a linear combination ofe₁,e₂, and e₃.

To see that this is so, take an arbitrary vector (a₁,a₂,a₃) inR³, and write:

{\begin{aligned}(a_{1},a_{2},a_{3})&=(a_{1},0,0)+(0,a_{2},0)+(0,0,a_{3})\\[6pt]&=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1)\\[6pt]&=a_{1}\mathbf {e} _{1}+a_{2}\mathbf {e} _{2}+a_{3}\mathbf {e} _{3}.\end{aligned}}

Functions

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LetK be the setC of allcomplex numbers, and letV be the set C_C(R) of allcontinuous functions from thereal lineR to thecomplex planeC.Consider the vectors (functions)f andg defined byf(t) :=e^it andg(t) :=e^−it.(Here,e is thebase of the natural logarithm, about 2.71828..., andi is theimaginary unit, a square root of −1.)Some linear combinations off andg are:

$\cos t={\tfrac {1}{2}}\,e^{it}+{\tfrac {1}{2}}\,e^{-it}$
$2\sin t=(-i)e^{it}+(i)e^{-it}.$

On the other hand, the constant function 3 isnot a linear combination off andg. To see this, suppose that 3 could be written as a linear combination ofe^it ande^−it. This means that there would exist complex scalarsa andb such thatae^it +be^−it = 3 for all real numberst. Settingt = 0 andt = π gives the equationsa +b = 3 anda +b = −3, and clearly this cannot happen.(SeeEuler's identity.)

Polynomials

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LetK beR,C, or any field, and letV be the setP of allpolynomials with coefficients taken from the fieldK.Consider the vectors (polynomials)p₁ := 1,p₂ :=x + 1, andp₃ :=x² +x + 1.

To find out whetherx² − 1 is a linear combination ofp₁,p₂, andp₃,we consider an arbitrary linear combination of these vectors and try to see when it equals the desired vectorx² − 1.Picking arbitrary coefficientsa₁,a₂, anda₃, 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 ofx, we get

a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})=1x^{2}+0x+(-1).

Two polynomials are equalif 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.

Thissystem of linear equations can easily be solved.First, the first equation simply says thata₃ is 1.Knowing that, we can solve the second equation fora₂, which comes out to −1.Finally, the last equation tells us thata₁ 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}

sox² − 1is a linear combination ofp₁,p₂, and p₃.

On the other hand, if we try to makex³ − 1 a linear combination ofp₁,p₂, andp₃, then following the same process as before, we get the equation

{\begin{aligned}&0x^{3}+a_{3}x^{2}+(a_{2}+a_{3})x+(a_{1}+a_{2}+a_{3})\\[5pt]={}&1x^{3}+0x^{2}+0x+(-1).\end{aligned}}

However, when we set corresponding coefficients equal in this case, the equation forx³ is

0=1

which is always false.Therefore, there is no way for this to work, andx³ − 1 isnot a linear combination ofp₁,p₂, and p₃.

The linear span

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Main article:Linear span

Take an arbitrary fieldK, an arbitrary vector spaceV, and letv₁,...,v_n be vectors (inV).It is interesting to consider the set ofall linear combinations of these vectors.This set is called thelinear span (or justspan) of the vectors, sayS = {v₁, ...,v_n}. We write the span ofS as span(S)^[4]^[5] or sp(S):

\operatorname {span} (\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}):=\{a_{1}\mathbf {v} _{1}+\cdots +a_{n}\mathbf {v} _{n}:a_{1},\ldots ,a_{n}\in K\}.

Linear independence

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Main article:Linear independence

Suppose that, for some sets of vectorsv₁,...,v_n,a single vector can be written in two different ways as a linear combination of them:

\mathbf {v} =\sum _{i}a_{i}\mathbf {v} _{i}=\sum _{i}b_{i}\mathbf {v} _{i}{\text{ where }}a_{i}\neq b_{i}.

This is equivalent, by subtracting these ( $c_{i}:=a_{i}-b_{i}$ ), to saying a non-trivial combination is zero:^[6]^[7]

\mathbf {0} =\sum _{i}c_{i}\mathbf {v} _{i}.

If that is possible, thenv₁,...,v_n are calledlinearly dependent; otherwise, they arelinearly independent.Similarly, we can speak of linear dependence or independence of an arbitrary setS of vectors.

IfS is linearly independent and the span ofS equalsV, thenS is abasis forV.

Affine, conical, and convex combinations

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By restricting the coefficients used in linear combinations, one can define the related concepts ofaffine combination,conical combination, andconvex combination, and the associated notions of sets closed under these operations.

Type of combination	Restrictions on coefficients	Name of set	Model space
Linear combination	no restrictions	Vector subspace	$\mathbf {R} ^{n}$
Affine combination	${\textstyle \sum a_{i}=1}$	Affine subspace	Affinehyperplane
Conical combination	$a_{i}\geq 0$	Convex cone	Quadrant,octant, ororthant
Convex combination	$a_{i}\geq 0$ and ${\textstyle \sum a_{i}=1}$	Convex set	Simplex

Because these are morerestricted operations, more subsets will be closed under them, so affine subsets, convex cones, and convex sets aregeneralizations of vector subspaces: a vector subspace is also an affine subspace, a convex cone, and a convex set, but a convex set need not be a vector subspace, affine, or a convex cone.

These concepts often arise when one can take certain linear combinations of objects, but not any: for example,probability distributions are closed under convex combination (they form a convex set), but not conical or affine combinations (or linear), andpositive measures are closed under conical combination but not affine or linear – hence one definessigned measures as the linear closure.

Linear and affine combinations can be defined over any field (or ring), but conical and convex combination require a notion of "positive", and hence can only be defined over anordered field (orordered ring), generally the real numbers.

If one allows onlyscalar multiplication, not addition, one obtains a (not necessarily convex)cone; one often restricts the definition to only allowing multiplication by positive scalars.

All of these concepts are usually defined as subsets of an ambient vector space (except for affine spaces, which are also considered as "vector spaces forgetting the origin"), rather than being axiomatized independently.

Operad theory

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Main article:Operad theory

More abstractly, in the language ofoperad theory, one can consider vector spaces to bealgebras over the operad $\mathbf {R} ^{\infty }$ (the infinitedirect sum, so only finitely many terms are non-zero; this corresponds to only taking finite sums), which parametrizes linear combinations: the vector $(2,3,-5,0,\dots )$ for instance corresponds to the linear combination $2\mathbf {v} _{1}+3\mathbf {v} _{2}-5\mathbf {v} _{3}+0\mathbf {v} _{4}+\cdots$ . Similarly, one can consider affine combinations, conical combinations, and convex combinations to correspond to the sub-operads where the terms sum to 1, the terms are all non-negative, or both, respectively. Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex. This formalizes what is meant by $\mathbf {R} ^{n}$ being or the standard simplex being model spaces, and such observations as that every boundedconvex polytope is the image of a simplex. Here suboperads correspond to more restricted operations and thus more general theories.

From this point of view, we can think of linear combinations as the most general sort of operation on a vector space – saying that a vector space is an algebra over the operad of linear combinations is precisely the statement thatall possible algebraic operations in a vector space are linear combinations.

The basic operations of addition and scalar multiplication, together with the existence of anadditive identity and additive inverses, cannot be combined in any more complicated way than the generic linear combination: the basic operations are agenerating set for the operad of all linear combinations.

Ultimately, this fact lies at the heart of the usefulness of linear combinations in the study of vector spaces.

Generalizations

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IfV is atopological vector space, then there may be a way to make sense of certaininfinite linear combinations, using the topology ofV.For example, we might be able to speak ofa₁v₁ +a₂v₂ +a₃v₃ + ⋯, going on forever.Such infinite linear combinations do not always make sense; we call themconvergent 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 flavors of topological vector spaces go into more detail about these.

IfK is acommutative ring instead of a field, then everything that has been said above about linear combinations generalizes to this case without change.The only difference is that we call spaces like thisVmodules instead of vector spaces.IfK is anoncommutative ring, then the concept still generalizes, 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 whenV is abimodule over two rings,K_L andK_R.In that case, the most general linear combination looks like

a_{1}\mathbf {v} _{1}b_{1}+\cdots +a_{n}\mathbf {v} _{n}b_{n}

wherea₁,...,a_n belong toK_L,b₁,...,b_n belong toK_R, andv₁,…,v_n belong toV.

Citations

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^Strang (2016) p. 3, § 1.1
^Lay, Lay & McDonald (2016) p. 28, ch. 1
^Axler (2015) p. 28, § 2.3
^Axler (2015) pp. 29-30, §§ 2.5, 2.8
^Katznelson & Katznelson (2008) p. 9, § 1.2.3
^Axler (2015) pp. 32-33, §§ 2.17, 2.19
^Katznelson & Katznelson (2008) p. 14, § 1.3.2

References

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Axler, Sheldon Jay (2015).Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer.doi:10.1007/978-3-319-11080-6.ISBN 978-3-319-11079-0.
Katznelson, Yitzhak; Katznelson, Yonatan R. (2008).A (Terse) Introduction to Linear Algebra.American Mathematical Society.ISBN 978-0-8218-4419-9.
Lay, David C.; Lay, Steven R.; McDonald, Judi J. (2016).Linear Algebra and its Applications (5th ed.). Pearson.ISBN 978-0-321-98238-4.
Strang, Gilbert (2016).Introduction to Linear Algebra (5th ed.). Wellesley Cambridge Press.ISBN 978-0-9802327-7-6.

External links

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Linear Combinations and Span: Understanding linear combinations and spans of vectors, khanacademy.org.

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