Linear Transformation

  • Thread starter hackett5
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  • #1
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Homework Statement


Define a Function T : P3 → ℝ3 by
T(p) = [p(3), p'(1), 01 p(x) dx ]

Show that T is a linear transformation

Homework Equations


From the definition of a linear transformation:
f(v1 + v2) = f(v1) + f(v2)
and
f(cv) = cf(v)


The Attempt at a Solution


This is how I've started the problem, but I'm not sure I'm heading in the right direction. Either way I'm stuck.

T(p+q) = (p+q)(3), (p+q)'(1), ∫(p+q)(x)dx
= p(3) + q(3), p'(1) + q'(1), ∫p(x)dx + ∫q(x)dx

I'm not sure where to go from here. Thanks

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
LCKurtz
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Homework Statement


Define a Function T : P3 → ℝ3 by
T(p) = [p(3), p'(1), 01 p(x) dx ]

Show that T is a linear transformation

Homework Equations


From the definition of a linear transformation:
f(v1 + v2) = f(v1) + f(v2)
and
f(cv) = cf(v)


The Attempt at a Solution


This is how I've started the problem, but I'm not sure I'm heading in the right direction. Either way I'm stuck.

T(p+q) = (p+q)(3), (p+q)'(1), ∫(p+q)(x)dx
= [p(3) + q(3), p'(1) + q'(1), ∫p(x)dx + ∫q(x)dx]

I'm not sure where to go from here.

Just keep going. Does that = T(p)+T(q)? (And remember, those are definite integrals; no constants of integration there).
 

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