lim f (x) : ;-.

#x#;-

By this is meant that for each M 9 0 there exists an R 9 0 such that if #x# . R,

then f (x) 9 M. Such a function f is said to be coercive. Show that if f is

continuous on RL and is coercive, then f attains a minimum on RL.

LINEAR TRANSFORMATIONS 21

10. LINEAR TRANSFORMATIONS

A linear transformation from RL to RK is a function T with domain RL and

range contained in RK such that for every x, y in RL and every pair of scalars

,

T ( x ; y) : T (x) ; T (y).

Example 10.1. T : R ; R: Rotate each vector in R through an angle .

Example 10.2. T : R ; R de¬ned by

T (x , x , x ) : (x , x , 0).

This is the projection of R onto the x x plane.

Example 10.3. Let A be an m ; n matrix. De¬ne

T x : Ax. (1)

In the event that the range space is R, that is, we have a linear transfor-

mation from RL to R, we call the transformation a linear functional. The

following is an example of a linear functional L . Let a be a ¬xed vector in RL.

Then

L (x) : 1a, x2 (2)

de¬nes a linear functional.

We now show that (1) and (2) are essentially the only examples of a

linear transformation and linear functional, respectively. Recall that the

standard basis in RL is the basis consisting of the vectors e , . . . , e , where

L

e : (0, 0, . . . , 0, 1, 0, . . . , 0) and 1 is the ith entry.

G

T 10.1. L et T be a linear transformation from RL to RK. T hen relative to

the standard bases in RL and RK there exists an m ; n matrix A such that

T x : Ax.

Let x + RL ; then x : L x e . Hence

H H H

L L

T x:T x e : x (T e ). (3)

HH H H

H H

For each j : 1, . . . , n, T e is a vector in RK, so if e*, . . . , e* is the standard basis

K

H

in RK,

K

T e : a e*. (4)

GH G

H

G

22 TOPICS IN REAL ANALYSIS

Substituting (4) into (3) and changing the order of summation give

L K K L

T x : x a e* : a x e*.

GH G GH H G

H

H G G H

Therefore if A is the m ; n matrix whose jth column is (a . . . a )R, then T x

H KH

expressed as an m-vector (y , . . . , y )R is given by

K

y

$ : T x : Ax.

y

K

From (4) we get that the jth column of A gives the coordinates of T e

H

relative to e*, . . . , e* .

G K

In the case of a linear functional L , the matrix A reduces to a row vector a

in RL, so that

L (x) : 1a, x2.

Exercise 10.1. Show that a linear functional L is a continuous mapping from

RL to R .

11. DIFFERENTIATION IN RL

Let D be an open interval in R and let f be real-valued function de¬ned on

D. The function f is said to have a derivative or to be differentiable at a point

x in D if

f (x ; h) 9 f (x )

(1)

lim

h

F

exists. The limit is called the derivative of f at x and is denoted by f (x ).

The question now arises as to how one generalizes this concept to real-

valued functions de¬ned on open sets D in RL and to functions de¬ned on open

sets in RL with range in RK. In elementary calculus the notion of partial

derivative for real-valued functions de¬ned on open sets D in RL is introduced.

For each i : 1, . . . , n the partial derivative with respect to x at a point x is

G

de¬ned by

*f f (x , . . . , x , x ; h, x , . . . , x ) 9 f (x , . . . , x )

G\ G G> L L (2)

(x ) : lim

*x h

G F

provided the limit on the right exists. It turns out that the notion of partial

derivative is not the correct generalization of the notion of derivative.

DIFFERENTIATION IN RL 23

Figure 1.1.

To motivate the correct generalization of the notion of derivative to

functions with domain and range in spaces of dimension greater than 1, we

reexamine the notion of derivative for functions f de¬ned on an open interval

D in R with range in R.

We rewrite (1) as

f (x ; h) 9 f (x )

; (h),

f (x ) :

h

where (h) ; 0 as h ; 0. This in turn can be rewritten as

f (x ; h) 9 f (x ) : f (x )h ; (h), (3)

where (h)/h ; 0 as h ; 0. The term f (x )h is a linear approximation to f near

x and it is a ˜˜good approximation for small h™™ in the sense that (h)/h ; 0 as

h ; 0. See Figure 1.1.

If we consider h to be an arbitrary real number, then f (x )h de¬nes a linear

functional L on R by the formula L (h) : f (x )h. Thus if f is differentiable at

x , there exists a linear functional (or linear transformation) on R such that

f (x ; h) 9 f (x ) : L (h) ; (h), (4)

where (h)/h ; 0 as h ; 0. Conversely, let there exist a linear functional L on

R such that (4) holds. Then L (h) : ah for some real number a, and we may

write

f (x ; h) 9 f (x ) : ah ; (h).

24 TOPICS IN REAL ANALYSIS

If we divide by h " 0 and then let h ; 0, we get that f (x ) exists and equals

a. Thus we could have used (4) to de¬ne the notion of derivative and could

have de¬ned the derivative to be the linear functional L , which in this case is

determined by the number a. The reader might feel that this is a very

convoluted way of de¬ning the notion of derivative. It has the merit, however,

of being the de¬nition that generalizes correctly to functions with domain in

RL and range in RK for any n . 1 and any m . 1.

Let D be an open set in RL, n . 1, and let f : ( f , . . . , f ) be a function

K

de¬ned on D with range in RK, m . 1. The function f is said to be differentiable

at a point x in D, or to be Frechet differentiable at x , if there exists a linear

´´

transformation T (x ) from RL to RK such that for all h in RL with h suf¬ciently

small

f(x ; h) 9 f(x ) : T (x )h ; (h), (5)

where # (h)#/#h# ; 0 as #h# ; 0.

We now represent T (x ) as a matrix relative to the standard bases e , . . . , e

L

in RL and e*, . . . , e* in RK. Let be a real number and let h : e for a ¬xed j

K H

in the set 1, . . . , n. Then e : (0, . . . , 0, , 0, . . . , 0), where the occurs in the jth

H

component. From (5) we have

f(x ; e ) 9 f(x ) : T (x )( e ) ; ( e ).

H H H

Hence for " 0

f(x ; e ) 9 f(x ) (e)

H : T (x )e ; H.

H

Since # e # : " " and for i : 1, . . . , m,

H

f (x ; e )9 f (x ): f (x , . . . , x ,x ; ,x ,...,x )

H\ H H> L

G H G G

9 f (x , . . . , x ),

L

it follows on letting ; 0 that for i : 1, . . . , m the partial derivatives *f (x )/*x

G H

exist and

*f

(x )

*x

H

T (x )e : $ .

H

*f

K (x )

*x

H

DIFFERENTIATION IN RL 25

Since the coordinates of T (x )e relative to the standard basis in RK are given

H

by the jth column of the matrix representing T (x ), the matrix representing

T (x ) is the matrix

*f

G (x ) .

*x

H

If we de¬ne

f to be the row vector

G

*f *f

G ,..., G ,

f : i : 1, . . . , n,

G *x *x

L

and

f to be the matrix

f

$,

f

K

then we have shown that

T (x ) :