Chapter 6 Application: Differential Operator is a Linear Operator
Let V be the vector space of infinitely differentiable functions on R. Then differentiation
may be thought of as a transformation that transforms functions to functions (from V into
V).
For f  V we define D ( f ) 
df
.
dx
Concrete Example
Let f ( x)  e x and let g ( x)  x 2 . Show by direct computation that
D(2 f  3g )  2 D( f )  3D( g ).
D(2 f  3g )  D(2e x  3x 2 )  2e x  6 x
 
2D( f )  3D( g )  2 e x  32 x   2e x  6 x
Proof
1. T u  v  T u   T (v)
“The Sum Rule”: D f x  g x  f x  g x
Let F ( x)  f ( x)  g ( x) .
F ( x  h)  F ( x )
f ( x  h)  g ( x  h)   f ( x )  g ( x ) 
F ( x)  lim
 lim
h 0
h 0
h
h
f ( x  h)  f ( x) fg ( x  h)  g ( x)
 lim

 f x   g x 
h 0
h
h
2. T cu   cT u 
“The Constant Multiple Rule”:
If c is a constant and f is a differentiable function, then D(cf )  cD f  .
Let g ( x)  c f x .
g ( x  h)  g ( x )
cf ( x  h)  cf ( x)
f ( x  h)  f ( x )
g ( x)  lim
 lim
 c lim
 cf x 
h 0
h 0
h 0
h
h
h
Example: Matrix Representation of Differentiation
Let P3 be the vector space of polynomials of degree 3.
The standard basis for this space is: 1, x, x 2 , x 3 which in vector notation is represented as


1 0 0 0 
        
0 1 0 0 
 , , , .
0 0 1 0 
0 0 0 1 
Construct the matrix M D that induces the transformation D : P3 t   P3 t  that sends
every polynomial p to its derivative p  . Use the transformation matrix to calculate the
derivative of pt   2  3t  5t 3 .
D1  0 corresponds to vector 0 0
Dx  1 corresponds to vector 1 0
Dx 2   2 x corresponds to vector 0
Dx 3   3x 2 corresponds to vector 0
Therefore, M D
0
0

0

0
0
0
2
0
0
0
0 0
3 0
1 0 0
0 2 0
.
0 0 3

0 0 0
Find the image of 2  3 0 5:
0
0

0

0
1 0 0  2   3
0 2 0  3  0 


.
0 0 3  0   15 
    
0 0 0  5   0 
So the derivative of pt   2  3t  5t 3 is  3  15t 2 .
Example: Matrix Representation of Differentiation
e , te , t e 
A  e , te , t e 
Let S be the vector space of functions spanned by the linearly independent
t
t
2
functions
t
t
t
. Thus, a basis for S is given by
2 t
. Let D : S  S be the map that sends a function in S to its
derivative. We can construct M D, A, A (that is, the matrix that induces D in terms of
coordinate basis A to itself).
0 A
 
Dte   e  te corresponds to vector 1 1 0A
Dt e   2te  t e corresponds to vector 0 2 1A
D e t  e t corresponds to coordinate vector 1 0
t
2
t
t
t
t
2
t
1 1 0
M D, A, A  0 1 2 .
0 0 1
Find the derivative of f t   2e t  3te t  t 2 e t .
1 1 0  2   1
0 1 2   3   1 .

    
0 0 1  1   1 
Thus the derivative is  e t  te t  t 2 e t .
Example: Solving a Differential Equation
Let L : P2  P2 be the linear transformation defined by L y   x 2 y   y   y .
Find the matrix M of L with respect to the standard basis B of P2. Use your answer to find
all solutions y in P2 to the following differential equation:
x 2 y   y   y  3  3x  x 2 .
1 
0  :
 
0 
What is the image of px   1 ? L y   x  0  0  1  1
2
1 
0 
 
0 
0 
1  :
 
0 
What is the image of px  x ? L y   x 2  0  1  x  1  x
0 
0  :
 
1 
What is the image of px   x ? L y   x  2  2 x  x  2 x  3x
2
2
3
1  1 0 


Thus M L  0 1  2 and we want to solve M L x  3 .
1
0 0
3 


We row reduce  M L

3  1  1 0 3 
 

| 3   0 1  2 3 .
1  0 0
3 1 
 1  1 0 3   1 0 0 20 
3

 
RREF  0 1  2 3   0 1 0 11 3 .
 

3 1   0 0 1 1 
 0 0
3 

And linear algebra tells us the solution is unique in P2!
2
 1
1
 
 0 
2
0
 2
 
 3 