Page 115 - 35Linear Algebra

6.3 Linear Differential Operators 115

line is 1 dimensional, but the careful definition of “dimension” takes some work; this
is tackled in Chapter 11.) This leads us to write
          
1 0 0 0 0 0
c 1
1
1
1
1
L c 1   + c 2   = c 1   + c 2   =  1 1  .

c 2
0 1 0 0 0 0
 
0 0
This makes sense, but requires a warning: The matrix  1 1  specifies L so long
0 0
as you also provide the information that you are labeling points in the plane V by the
two numbers (c 1 , c 2 ).
6.3 Linear Differential Operators

Your calculus class became much easier when you stopped using the limit
definition of the derivative, learned the power rule, and started using linearity
of the derivative operator.

Example 72 Let V be the vector space of polynomials of degree 2 or less with standard
addition and scalar multiplication;
2
V := {a 0 · 1 + a 1 x + a 2 x | a 0 , a 1 , a 2 ∈ R}
Let d : V → V be the derivative operator. The following three equations, along with
dx
linearity of the derivative operator, allow one to take the derivative of any 2nd degree
polynomial:
d d d 2
1 = 0, x = 1, x = 2x .
dx dx dx
In particular

d d d d
2 2
(a 0 1 + a 1 x + a 2 x ) = a 0 1 + a 1 x + a 2 x = 0 + a 1 + 2a 2 x.
dx dx dx dx
Thus, the derivative acting any of the infinitely many second order polynomials is
determined by its action for just three inputs.

6.4 Bases (Take 1)

The central idea of linear algebra is to exploit the hidden simplicity of linear
functions. It ends up there is a lot of freedom in how to do this. That
freedom is what makes linear algebra powerful.

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