Define T: P3-->P3 by T(f)(t)=2f(t)+(1-t)f'(t)
Show that T is a linear transformation.
I've included the problem that I'm working on. I'm having a hard time proving this transformation is linear because of the (1-t)f'(t) term at the end of it. I'm not entirely sure what to do with it when I evaluate T(s+t)=T(s)+T(t). Do I make it (1-t+s)f'(t)?
Also looking ahead, I'm not sure how to approach part b) as he skipped that in lecture.
For part b) that was how i started the first few times i attempted this problem. I'm just not sure what to do with the T(t) term that equals 2t+(1-t) and the T(t^3)=2t^3+2t^2-3t^5. I put in in like this, but I think its not correct, as I'm not sure what happens with the (1-t) term and the t^5 term in this situation:
[2 (1-t) 0 0
0 2 -2 0
0 0 2 3
0 0 -2 2]
no, that is nowhere near correct. you are not evaluating T correctly. we can write an element of P3 as: f(t) = a + bt + ct^{2} + dt^{3}
(P3 is a 4-dimensional space, and in the basis B = {1,t,t^{2},t^{3}}, we have [f]_{B} = [a,b,c,d]_{B}).
using the definition of T, we have: T(f(t)) = 2f(t) + (1-t)f'(t). thus:
T(f(t)) = T(a + bt + ct^{2}+dt^{3}) = 2(a + bt + ct^{2} + dt^{3}) + (1 - t)(b + 2ct + 3dt^{2})
= 2a + 2bt + 2ct^{2} + 2dt^{3} + b + 2ct + 3dt^{2} - bt - 2ct^{2} - 3dt^{3}
= (2a + b) + (2b + 2c - b)t + (2c + 3d - 2c)t^{2} + (2d - 3d)t^{3}
= (2a + b) + (b + 2c)t + 3dt^{2} - 3dt^{3} <---no powers of t higher than 3.
for the function f(t) = 1 (a constant function), a = 1, b = c = d = 0, so
T(1) = 2.
for the function f(t) = t, we have a = 0, b = 1, c = d = 0, so
T(t) = 1 + t
for the function f(t) = t^{2}, we have a = b = 0, c = 1, d = 0, so:
T(t^{2}) = 2t
for the function f(t) = t^{3}, we have a = b = c = 0, d = 1, so
T(t^{3}) = 3t^{2} - 3t^{3}
you're thinking of "t" as something that T acts on. it's not. it's what you call a "dummy variable", so that we don't have to write f as something like:
f = (constant1)*(constant function 1) + (constant2)*(identity function) + (constant3)*(squaring function) + (constant4)*(cubing function).
we could easily think of T as the function that goes from R^{4} to R^{4}:
T(a,b,c,d) = (2a+b,b+2c,3d,-3d), or more properly:
[T]_{B}([a,b,c,d]_{B}) = [2a+b,b+2c,3d,-3d]_{B}
since P3 is isomorphic to R^{4} using the isomorphism:
(1,0,0,0) → 1
(0,1,0,0) → t
(0,0,1,0) → t^{2}
(0,0,0,1) → t^{3}
in other words, you need to start thinking of a polynomial in t as being "a vector consisting of its coefficients".