Linear Transformation: Proving Linearity with Function T : P3 → ℝ3

[hackett5]

Homework Statement

Define a Function T : P₃ → ℝ³ by
T(p) = [p(3), p'(1), ₀∫¹ 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

 

[LCKurtz]

Homework Statement

Define a Function T : P₃ → ℝ³ by
T(p) = [p(3), p'(1), ₀∫¹ 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).