In mathematics, a linear combination is an expression constructed from a set of
terms by multiplying each term by a constant and adding the results. The
concept of linear combinations is 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 generalizations given
at the end of the article. 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,...,vn are vectors
and a1,...,an are scalars, then the linear combination of those vectors with
those scalars as coefficients is 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, like in the assertion "the
set of all linear combinations of v1,...,vn always forms a subspace".
However, one could also say "two different linear combinations can have
the same value" in which case the expression must have been meant. 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 is uniquely so. In any case, even when
viewed as expressions, all that matters about a linear combination is the
coefficient of each vi; trivial modifications such as permuting the
terms or adding terms with zero coefficient do not give 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 v1,...,vn, with the
coefficients unspecified. 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. Finally, we
may speak simply of a linear combination, where nothing is specified;
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 finitely many
vectors. However, the set S that the vectors are taken from 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. Examples and counterexamples
= Euclidean vectors= 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 =, e2 = and e3 =. Then any vector in R3 is a linear combination of
e1, e2 and e3. To see that this is so, take an
arbitrary vector in R3, and write: = Functions=
Let K be the set C of all complex numbers, and let V be the set CC(R) of
all continuous functions from the real line R to the complex plane C. Consider
the vectors f and g defined by f(t) := eit and g(t) := e−it. Some linear
combinations of f and g are: 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 eit and e−it. This means that there would
exist complex scalars a and b such that aeit + 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. See Euler's identity.
= Polynomials= Let K be R, C, or any field, and let V
be the set P of all polynomials with coefficients taken from the field K.
Consider the vectors 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 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 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,
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 Take an arbitrary field K, an arbitrary
vector space V, and let v1,...,vn be vectors. It’s interesting to consider
the set of all linear combinations of these vectors. This set is called the
linear span of the vectors, say S ={v1,...,vn}. We write the span of S as
span(S) or sp(S): Linear independence
For some sets of vectors v1,...,vn, a single vector can be written in two
different ways as a linear combination of them:
Equivalently, by subtracting these a non-trivial combination is zero:
If that is possible, then v1,...,vn 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. Affine, conical, and convex combinations
By restricting the coefficients used in linear combinations, one can define the
related concepts of affine combination, conical combination, and convex
combination, and the associated notions of sets closed under these operations.
Because these are more restricted operations, more subsets will be closed
under them, so affine subsets, convex cones, and convex sets are
generalizations 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, but not conical or affine combinations, and
positive measures are closed under conical combination but not affine or
linear – hence one defines signed measures as the linear closure.
Linear and affine combinations can be defined over any field, but conical and
convex combination require a notion of "positive", and hence can only be
defined over an ordered field, generally the real numbers.
If one allows only scalar multiplication, not addition, one
obtains a 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, rather than being axiomatized
independently. Operad theory
More abstractly, in the language of operad theory, one can consider vector
spaces to be algebras over the operad , which parametrizes linear combinations:
the vector for instance corresponds to the linear combination . 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 being or the standard simplex being model
spaces, and such observations as that every bounded convex 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 that all possible algebraic operations in a
vector space are linear combinations. 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: the basic operations are a generating 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 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. 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 generalizes to this case without change.
The only difference is that we call spaces like this V modules instead of
vector spaces. If K is a noncommutative 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 when V is a bimodule over two rings, KL and KR. In
that case, the most general linear combination looks like
where a1,...,an belong to KL, b1,...,bn belong to KR, and v1,...,vn belong to V.
References External links
Linear Combinations and Span: Understanding linear combinations and
spans of vectors, khanacademy.org.