Any equation of the form ax + by = c represents a
LINE in two-dimensional space. (Solving for y, assuming that b is not zero, yields the linear function
y = – x + .) An equation of the form ax + by + cz
= d represents a PLANE in three-dimensional space.
A linear combination of variables x 1 , x 2 , x 3 , … is a
sum of the form
a 1 x 1 + a 2 x 2 + a 3 x 3 +…
for some constants a 1 , a 2 , a 3 , … In VECTOR SPACE theory, a set of vectors is said to be linearly dependent if
some linear combination of those vectors is zero.
In LINEAR ALGEBRA, a MATRIX equation of the form
Ax = b is called a linear equation. It represents a system
of SIMULTANEOUS LINEAR EQUATIONS.
A linear differential equation is a DIFFERENTIAL
EQUATION of the form:
for some constants a 0 , a 1 , a 2 , …, a n and some fixed
function f(x).
In some settings it is appropriate to apply the term
linear to specific variables appearing in a complicated
expression. For instance, the term 5x
2
yz is linear with
respect to y and with respect to z.
See also EQUATION OF A LINE; EQUATION OF A
PLANE; LINEAR TRANSFORMATION; LINEARLY DEPENDENT AND INDEPENDENT.
linearly dependent and independent A collection
of functions is said to be linearly dependent if one of
them can be expressed as a sum of constant multiples
of the other; if this is not possible, then the collection is
said to be linearly independent. For example, the functions f 1 (x) = x, f 2 (x) = x
2 – 2x, f 3 (x) = x
2 are linearly
dependent, since f 3 (x) = 2f 1 (x) + f 2 (x). The functions
{x, 7x} are also linearly dependent, since the second
function is a constant multiple of the first. On the other
hand, the functions {x, x
2
, x
3
} are linearly independent,
as are the functions {sin x, cos x}.
A set of VECTORS is said to be linearly dependent if
it is possible to write one vector as a combination of
the remaining vectors. Equivalently, vectors v 1 , v 2 ,…,v n
are linearly dependent if it is possible to choose scalars
c 1 ,c 2 ,…,c n , not all zero, so that
c 1 v 1 + c 2 v 2 +…+ c n v n = 0
(If c i , say, is not zero, then dividing through by this
scalar shows that v i is a sum of multiples of the
remaining vectors.) If this is not possible, then the vectors are said to be linearly independent. For example,
in three-dimensional space, the vectors i = <1,0,0>, j =
<0,1,0> and k = <0,0,1> are linearly independent—it is
not possible to write any one as a sum of multiples of
the other two.
A basis for a VECTOR SPACE is a collection of linearly independent vectors with the property that any
other vector in the vector space can be written as a
sum of multiples of these vectors. For example, the
vectors i, j, and k form a basis for the vector space of
three-dimensional vectors for any other vector a =
that can be expressed as the combination
a = a 1 i + a 2 j + a 3 k. It is impossible to express a vector
as a combination of basis vectors in two different
ways. (To explain: Suppose v 1 , v 2 , v 3 is a basis for a
vector space, and that some vector a can be expressed
as a combination of these vectors in two different
ways: a = a 1 v 1 + a 2 v 2 + a 3 v 3 = b 1 v 1 + b 2 v 2 + b 3 v 3 . Subtracting gives the equation (a 1 – b 1 )v 1 + (a 2 – b 2 )v 2 +
(a 3 – b 3 )v 3 = 0. Since the vectors v 1 ,v 2 ,v 3 are linearly
independent, it must be the case that a 1 = b 1 , a 2 = b 2 ,
and a 3 = b 3 .)
Mathematicians have proved that every vector
space must have a basis, and that the number of vectors in any basis for a particular vector space is always
the same. This number is called the dimension of the
vector space. In particular, the set of all functions is a
vector space and so must have a basis. One candidate
for such a basis is the infinite collection of functions
{1,x,x
2
,x
3
,x
4
,…}. This set is certainly linearly independent, and the work of constructing TAYLOR SERIES
shows that all “appropriately nice” functions can be
expressed as infinite sums of these basic functions.
Functions like sin(x), cos(x), and sin(7x) repeat values
every 2π and are called periodic. Mathematicians have
shown that the collection {1,sin(x), cos(x), sin(2x),
cos(2x), sin(3x), cos(3x),…} forms a basis for the vector space of all periodic functions. This leads to the
study of FOURIER SERIES.
See also ORTHOGONAL.
a y a
dy
dx
a
d y
dx
a
d y
dx
f x
n
n
n
0
1
2
2
2
+
+
+ +
=
L
( )
c
–
b
a
–
b
linearly dependent and independent 315
LINE in two-dimensional space. (Solving for y, assuming that b is not zero, yields the linear function
y = – x + .) An equation of the form ax + by + cz
= d represents a PLANE in three-dimensional space.
A linear combination of variables x 1 , x 2 , x 3 , … is a
sum of the form
a 1 x 1 + a 2 x 2 + a 3 x 3 +…
for some constants a 1 , a 2 , a 3 , … In VECTOR SPACE theory, a set of vectors is said to be linearly dependent if
some linear combination of those vectors is zero.
In LINEAR ALGEBRA, a MATRIX equation of the form
Ax = b is called a linear equation. It represents a system
of SIMULTANEOUS LINEAR EQUATIONS.
A linear differential equation is a DIFFERENTIAL
EQUATION of the form:
for some constants a 0 , a 1 , a 2 , …, a n and some fixed
function f(x).
In some settings it is appropriate to apply the term
linear to specific variables appearing in a complicated
expression. For instance, the term 5x
2
yz is linear with
respect to y and with respect to z.
See also EQUATION OF A LINE; EQUATION OF A
PLANE; LINEAR TRANSFORMATION; LINEARLY DEPENDENT AND INDEPENDENT.
linearly dependent and independent A collection
of functions is said to be linearly dependent if one of
them can be expressed as a sum of constant multiples
of the other; if this is not possible, then the collection is
said to be linearly independent. For example, the functions f 1 (x) = x, f 2 (x) = x
2 – 2x, f 3 (x) = x
2 are linearly
dependent, since f 3 (x) = 2f 1 (x) + f 2 (x). The functions
{x, 7x} are also linearly dependent, since the second
function is a constant multiple of the first. On the other
hand, the functions {x, x
2
, x
3
} are linearly independent,
as are the functions {sin x, cos x}.
A set of VECTORS is said to be linearly dependent if
it is possible to write one vector as a combination of
the remaining vectors. Equivalently, vectors v 1 , v 2 ,…,v n
are linearly dependent if it is possible to choose scalars
c 1 ,c 2 ,…,c n , not all zero, so that
c 1 v 1 + c 2 v 2 +…+ c n v n = 0
(If c i , say, is not zero, then dividing through by this
scalar shows that v i is a sum of multiples of the
remaining vectors.) If this is not possible, then the vectors are said to be linearly independent. For example,
in three-dimensional space, the vectors i = <1,0,0>, j =
<0,1,0> and k = <0,0,1> are linearly independent—it is
not possible to write any one as a sum of multiples of
the other two.
A basis for a VECTOR SPACE is a collection of linearly independent vectors with the property that any
other vector in the vector space can be written as a
sum of multiples of these vectors. For example, the
vectors i, j, and k form a basis for the vector space of
three-dimensional vectors for any other vector a =
that can be expressed as the combination
a = a 1 i + a 2 j + a 3 k. It is impossible to express a vector
as a combination of basis vectors in two different
ways. (To explain: Suppose v 1 , v 2 , v 3 is a basis for a
vector space, and that some vector a can be expressed
as a combination of these vectors in two different
ways: a = a 1 v 1 + a 2 v 2 + a 3 v 3 = b 1 v 1 + b 2 v 2 + b 3 v 3 . Subtracting gives the equation (a 1 – b 1 )v 1 + (a 2 – b 2 )v 2 +
(a 3 – b 3 )v 3 = 0. Since the vectors v 1 ,v 2 ,v 3 are linearly
independent, it must be the case that a 1 = b 1 , a 2 = b 2 ,
and a 3 = b 3 .)
Mathematicians have proved that every vector
space must have a basis, and that the number of vectors in any basis for a particular vector space is always
the same. This number is called the dimension of the
vector space. In particular, the set of all functions is a
vector space and so must have a basis. One candidate
for such a basis is the infinite collection of functions
{1,x,x
2
,x
3
,x
4
,…}. This set is certainly linearly independent, and the work of constructing TAYLOR SERIES
shows that all “appropriately nice” functions can be
expressed as infinite sums of these basic functions.
Functions like sin(x), cos(x), and sin(7x) repeat values
every 2π and are called periodic. Mathematicians have
shown that the collection {1,sin(x), cos(x), sin(2x),
cos(2x), sin(3x), cos(3x),…} forms a basis for the vector space of all periodic functions. This leads to the
study of FOURIER SERIES.
See also ORTHOGONAL.
a y a
dy
dx
a
d y
dx
a
d y
dx
f x
n
n
n
0
1
2
2
2
+
+
+ +
=
L
( )
c
–
b
a
–
b
linearly dependent and independent 315