![Vec-dep](https://upload.wikimedia.org/wikipedia/commons/a/ab/Vec-dep.png)
In the theory of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s is said to be if there is a nontrivial
linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are central to the definition of
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
.
A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space.
Definition
A sequence of vectors
from a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
is said to be ''linearly dependent'', if there exist
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
not all zero, such that
:
where
denotes the zero vector.
This implies that at least one of the scalars is nonzero, say
, and the above equation is able to be written as
:
if
and
if
Thus, a set of vectors is linearly dependent if and only if one of them is zero or a
linear combination of the others.
A sequence of vectors
is said to be ''linearly independent'' if it is not linearly dependent, that is, if the equation
:
can only be satisfied by
for
This implies that no vector in the sequence can be represented as a linear combination of the remaining vectors in the sequence. In other words, a sequence of vectors is linearly independent if the only representation of
as a linear combination of its vectors is the trivial representation in which all the scalars
are zero.
Even more concisely, a sequence of vectors is linearly independent if and only if
can be represented as a linear combination of its vectors in a unique way.
If a sequence of vectors contains the same vector twice, it is necessarily dependent. The linear dependency of a sequence of vectors does not depend of the order of the terms in the sequence. This allows defining linear independence for a finite set of vectors: A finite set of vectors is ''linearly independent'' if the sequence obtained by ordering them is linearly independent. In other words, one has the following result that is often useful.
A sequence of vectors is linearly independent if and only if it does not contain the same vector twice and the set of its vectors is linearly independent.
Infinite case
An infinite set of vectors is ''linearly independent'' if every nonempty finite
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
is linearly independent. Conversely, an infinite set of vectors is ''linearly dependent'' if it contains a finite subset that is linearly dependent, or equivalently, if some vector in the set is a linear combination of other vectors in the set.
An
indexed family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, wher ...
of vectors is ''linearly independent'' if it does not contain the same vector twice, and if the set of its vectors is linearly independent. Otherwise, the family is said ''linearly dependent''.
A set of vectors which is linearly independent and
spans some vector space, forms a
basis for that vector space. For example, the vector space of all
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s in over the reals has the (infinite) subset as a basis.
Geometric examples
![Vectores independientes](https://upload.wikimedia.org/wikipedia/commons/2/27/Vectores_independientes.png)
*
and
are independent and define the plane P.
*
,
and
are dependent because all three are contained in the same plane.
*
and
are dependent because they are parallel to each other.
*
,
and
are independent because
and
are independent of each other and
is not a linear combination of them or, equivalently, because they do not belong to a common plane. The three vectors define a three-dimensional space.
* The vectors
(null vector, whose components are equal to zero) and
are dependent since
Geographic location
A person describing the location of a certain place might say, "It is 3 miles north and 4 miles east of here." This is sufficient information to describe the location, because the geographic coordinate system may be considered as a 2-dimensional vector space (ignoring altitude and the curvature of the Earth's surface). The person might add, "The place is 5 miles northeast of here." This last statement is ''true'', but it is not necessary to find the location.
In this example the "3 miles north" vector and the "4 miles east" vector are linearly independent. That is to say, the north vector cannot be described in terms of the east vector, and vice versa. The third "5 miles northeast" vector is a
linear combination of the other two vectors, and it makes the set of vectors ''linearly dependent'', that is, one of the three vectors is unnecessary to define a specific location on a plane.
Also note that if altitude is not ignored, it becomes necessary to add a third vector to the linearly independent set. In general, linearly independent vectors are required to describe all locations in -dimensional space.
Evaluating linear independence
The zero vector
If one or more vectors from a given sequence of vectors
is the zero vector
then the vector
are necessarily linearly dependent (and consequently, they are not linearly independent).
To see why, suppose that
is an index (i.e. an element of
) such that
Then let
(alternatively, letting
be equal any other non-zero scalar will also work) and then let all other scalars be
(explicitly, this means that for any index
other than
(i.e. for
), let
so that consequently
).
Simplifying
gives:
:
Because not all scalars are zero (in particular,
), this proves that the vectors
are linearly dependent.
As a consequence, the zero vector can not possibly belong to any collection of vectors that is linearly ''in''dependent.
Now consider the special case where the sequence of
has length
(i.e. the case where
).
A collection of vectors that consists of exactly one vector is linearly dependent if and only if that vector is zero.
Explicitly, if
is any vector then the sequence
(which is a sequence of length
) is linearly dependent if and only if alternatively, the collection
is linearly independent if and only if
Linear dependence and independence of two vectors
This example considers the special case where there are exactly two vector
and
from some real or complex vector space.
The vectors
and
are linearly dependent
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
at least one of the following is true:
#
is a scalar multiple of
(explicitly, this means that there exists a scalar
such that
) or
#
is a scalar multiple of
(explicitly, this means that there exists a scalar
such that
).
If
then by setting
we have
(this equality holds no matter what the value of
is), which shows that (1) is true in this particular case. Similarly, if
then (2) is true because
If
(for instance, if they are both equal to the zero vector
) then ''both'' (1) and (2) are true (by using
for both).
If
then
is only possible if
''and''
; in this case, it is possible to multiply both sides by
to conclude
This shows that if
and
then (1) is true if and only if (2) is true; that is, in this particular case either both (1) and (2) are true (and the vectors are linearly dependent) or else both (1) and (2) are false (and the vectors are linearly ''in''dependent).
If
but instead
then at least one of
and
must be zero.
Moreover, if exactly one of
and
is
(while the other is non-zero) then exactly one of (1) and (2) is true (with the other being false).
The vectors
and
are linearly ''in''dependent if and only if
is not a scalar multiple of
''and''
is not a scalar multiple of
.
Vectors in R2
Three vectors: Consider the set of vectors
and
then the condition for linear dependence seeks a set of non-zero scalars, such that
:
or
:
Row reduce this matrix equation by subtracting the first row from the second to obtain,
:
Continue the row reduction by (i) dividing the second row by 5, and then (ii) multiplying by 3 and adding to the first row, that is
:
Rearranging this equation allows us to obtain
:
which shows that non-zero ''a''
''i'' exist such that
can be defined in terms of
and
Thus, the three vectors are linearly dependent.
Two vectors: Now consider the linear dependence of the two vectors
and
and check,
:
or
:
The same row reduction presented above yields,
:
This shows that
which means that the vectors ''v''
1 = (1, 1) and ''v''
2 = (−3, 2) are linearly independent.
Vectors in R4
In order to determine if the three vectors in
:
are linearly dependent, form the matrix equation,
:
Row reduce this equation to obtain,
:
Rearrange to solve for v
3 and obtain,
:
This equation is easily solved to define non-zero ''a''
i,
:
where
can be chosen arbitrarily. Thus, the vectors
and
are linearly dependent.
Alternative method using determinants
An alternative method relies on the fact that
vectors in
are linearly independent
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
formed by taking the vectors as its columns is non-zero.
In this case, the matrix formed by the vectors is
:
We may write a linear combination of the columns as
:
We are interested in whether for some nonzero vector Λ. This depends on the determinant of
, which is
:
Since the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
is non-zero, the vectors
and
are linearly independent.
Otherwise, suppose we have
vectors of
coordinates, with
Then ''A'' is an ''m''×''n'' matrix and Λ is a column vector with
entries, and we are again interested in ''A''Λ = 0. As we saw previously, this is equivalent to a list of
equations. Consider the first
rows of
, the first
equations; any solution of the full list of equations must also be true of the reduced list. In fact, if is any list of
rows, then the equation must be true for those rows.
:
Furthermore, the reverse is true. That is, we can test whether the
vectors are linearly dependent by testing whether
:
for all possible lists of
rows. (In case
, this requires only one determinant, as above. If
, then it is a theorem that the vectors must be linearly dependent.) This fact is valuable for theory; in practical calculations more efficient methods are available.
More vectors than dimensions
If there are more vectors than dimensions, the vectors are linearly dependent. This is illustrated in the example above of three vectors in
Natural basis vectors
Let
and consider the following elements in
, known as the
natural basis vectors:
:
Then
are linearly independent.
Linear independence of functions
Let
be the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of all differentiable
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
s of a real variable
. Then the functions
and
in
are linearly independent.
Proof
Suppose
and
are two real numbers such that
:
Take the first derivative of the above equation:
:
for values of
We need to show that
and
In order to do this, we subtract the first equation from the second, giving
. Since
is not zero for some
,
It follows that
too. Therefore, according to the definition of linear independence,
and
are linearly independent.
Space of linear dependencies
A linear dependency or
linear relation
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
More precisely, if e_1,\dots,e_n are elements of a (left) module over a ring ...
among vectors is a
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
with
scalar
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
components such that
:
If such a linear dependence exists with at least a nonzero component, then the vectors are linearly dependent. Linear dependencies among form a vector space.
If the vectors are expressed by their coordinates, then the linear dependencies are the solutions of a homogeneous
system of linear equations, with the coordinates of the vectors as coefficients. A
basis of the vector space of linear dependencies can therefore be computed by
Gaussian elimination
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
.
Generalizations
Affine independence
A set of vectors is said to be affinely dependent if at least one of the vectors in the set can be defined as an
affine combination In mathematics, an affine combination of is a linear combination
: \sum_^ = \alpha_ x_ + \alpha_ x_ + \cdots +\alpha_ x_,
such that
:\sum_^ =1.
Here, can be elements ( vectors) of a vector space over a field , and the coefficients \alpha_ ...
of the others. Otherwise, the set is called affinely independent. Any affine combination is a linear combination; therefore every affinely dependent set is linearly dependent. Conversely, every linearly independent set is affinely independent.
Consider a set of
vectors
of size
each, and consider the set of
augmented vectors
of size
each. The original vectors are affinely independent if and only if the augmented vectors are linearly independent.
Linearly independent vector subspaces
Two vector subspaces
and
of a vector space
are said to be if
More generally, a collection
of subspaces of
are said to be if
for every index
where
The vector space
is said to be a of
if these subspaces are linearly independent and
See also
*
References
*
External links
*
Linearly Dependent Functionsat WolframMathWorld.
on Linear Independence.
Introduction to Linear Independenceat KhanAcademy.
{{Matrix classes
Abstract algebra
Linear algebra
Articles containing proofs