c

–

b

a

–

b

linearly dependent and independent 315

Any equation of the form ax + by = crepresents a

LINE

in two-dimensional space. (Solving for y, assum-

ing that bis not zero, yields the linear function

y=– x+ .) An equation of the form ax + by + cz

= drepresents a

PLANE

in three-dimensional space.

A linear combination of variables x1, x2, x3, … is a

sum of the form

a1x1+ a2x2+ a3x3+…

for some constants a1, a2, a3, … In

VECTOR SPACE

the-

ory, 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= bis 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 a0, a1, a2, …, anand 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 5x2yz is linear with

respect to yand with respect to z.

See also

EQUATION OF A LINE

;

EQUATION OF A

PLANE

;

LINEAR TRANSFORMATION

;

LINEARLY DEPEN

-

DENT 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 func-

tions f1(x) = x, f2(x) = x2– 2x, f3(x) = x2are linearly

dependent, since f3(x) = 2f1(x) + f2(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, x2, x3} 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 v1, v2,…,vn

are linearly dependent if it is possible to choose scalars

c1,c2,…,cn, not all zero, so that

c1v1+ c2v2+…+ cnvn= 0

(If ci, say, is not zero, then dividing through by this

scalar shows that viis a sum of multiples of the

remaining vectors.) If this is not possible, then the vec-

tors 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 lin-

early 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 kform a basis for the vector space of

three-dimensional vectors for any other vector a=

<a1,a2,a3> that can be expressed as the combination

a= a1i+ a2j+ a3k. It is impossible to express a vector

as a combination of basis vectors in two different

ways. (To explain: Suppose v1, v2, v3is a basis for a

vector space, and that some vector acan be expressed

as a combination of these vectors in two different

ways: a= a1v1+ a2v2+ a3v3= b1v1+ b2v2+ b3v3. Sub-

tracting gives the equation (a1– b1)v1+ (a2– b2)v2+

(a3– b3)v3= 0. Since the vectors v1,v2,v3are linearly

independent, it must be the case that a1= b1, a2= b2,

and a3= b3.)

Mathematicians have proved that every vector

space must have a basis, and that the number of vec-

tors 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,x2,x3,x4,…}. This set is certainly linearly indepen-

dent, and the work of constructing T

AYLOR 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 2πand 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 vec-

tor space of all periodic functions. This leads to the

study of F

OURIER SERIES

.

See also

ORTHOGONAL

.