258 image

dy

––

dx

y

dy

––

dx

ORDER OF A MATRIX

is n. For example, the 3 ×3 iden-

tity matrix is:

If Ais a matrix with mrows and ncolumns, then

matrix multiplication shows we have:

AIn= A

ImA = A

that is, multiplication with the appropriate identity

matrix leaves any other matrix unchanged. If all the

matrices under consideration have the same number of

rows as columns, say nof each, then Inacts as an

IDEN

-

TITY ELEMENT

for that set under matrix multiplication.

Any 2 ×2 matrix Aof the form:

or

satisfies A2= I2. Thus any such matrix can be thought

of as a

SQUARE ROOT

of the 2 ×2 identity matrix. If

one is willing to permit the use of

COMPLEX NUMBERS

,

then the following matrix is an example of a cube root

of the 3 ×3 identity matrix:

This matrix satisfies the relation A3= I3.

image (range) The set of all values that a

FUNCTION

could adopt is called the image of the function. For

example, the image of the function y= x2defined for

all

REAL NUMBERS

is the set of all numbers greater than

or equal to zero. The term is also used for the output of

a specific input for the function. For instance, in the

example above, the image of the number 3 is 9.

In dealing with functions of real numbers, the term

range is preferred over image. Often mathematicians

will use the word image only when thinking of a prob-

lem geometrically. For example, if the function is a

GEOMETRIC TRANSFORMATION

such as a reflection in a

line, then one would speak of the image of geometric

figures under this transformation. In this example, the

image of any circle is another circle, and the image of a

straight line is another straight line.

implication See

CONDITIONAL

.

implicit differentiation When two variables xand y

satisfy a single equation F(x,y) = 0, it may be possible

to regard the equation nonetheless as defining yas a

function of x, even though no explicit formula of this

type may be apparent. (See

IMPLICIT FUNCTION

.) In

such a case, one can go further and differentiate the

equation as a whole, regarding yas a function of xand

using the

CHAIN RULE

in the process to find a formula

for the derivative . This process is known as

implicit differentiation. For example, if xy3+ 7x2y= 1,

then differentiating yields:

and so:

(assuming the denominator is not zero).

Implicit differentiation is useful for finding the

derivatives of inverse functions, for example. For

instance, if y= sin–1(x),then sin(y) = x. Differentiating

yields cos(y) = 1, and so,

.

Sometimes a function is more easily differentiated

if one first applies a logarithm and then differentiates