CLASS NOTES FOR LINEAR MATH

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CLASS NOTES FOR LINEAR MATH

• Section 3.7
• THE ALGEBRA OF LINEAR TRANSFORMATIONS

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COROLLARYS OF THE DEFINITIONS

• If T and S are linear transformations so is (TS)
  and (kT) where k is a scalor.
• The space ?L(V, W) of linear transformations TV
  ? W with the above and forms a vector space
• The resultant of T ? L(V,W) and S ? L(W,U) is
• T(S) ? L(V,U)
• 3. Composition is associative, and linear so the
  resultant of two linear transformations is linear.

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THE DERIVATIVE AS A LINEAR TRANSFORMATION
Multiplication by a constant (or even by a
function) is a linear transformation on F, the
vector space of functions. Let f and g be in F.
k(f g) kf kg (where is the addition of
F). Also, if f ? F Then kf ? F simply because F
is a vector space.
The derivative behaves this way as well
And
They are both linear transformations on F to F
and so the sum is also a linear transformation
(remember, the collection of linear
transformations from one vector space i.e. F to
another i.e. F is a vector space).
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LINEAR OPERATORS
In this way, the differential equation
can be written as
and since the derivative of the derivative is the
composition of linear transformations (otherwise
known as linear operators) the differential
equation
can be written in the form
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THE DERIVATIVE AS A LINEAR TRANSFORMATION
Suppose we wish to solve the differential equation
The solution now reduces to factoring the
associated polynomial
The equation can now be written as
Set u (D3)y
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THE DERIVATIVE AS A LINEAR TRANSFORMATION
Suppose we wish to solve the differential equation
Set u (D3)y
24x2 e-4x 48x
-e-x/4 - 48 e-4x/16 0
-e-4x/32
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THE DERIVATIVE AS A LINEAR TRANSFORMATION
Suppose we wish to solve the differential equation
Set u (D3)y
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THE DERIVATIVE AS A LINEAR TRANSFORMATION
Suppose we wish to solve the differential equation
AND THEN A CALCULATOR OCCURES!!!
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THE DERIVATIVE AS A LINEAR TRANSFORMATION
Suppose we wish to solve the differential equation
ANSWER
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EXAMPLE
Write D4 - 5D3 - D 5 as the resultant of
simpler operators
D4 - 5D3 - D 5 (D4 - 5D3 ) (D 5)
D3(D 5) (D 5)
(D3 1)(D 5)
(D 1)(D2 D 1)(D 5)
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End Section 3.7