Question

linear algebra: linear transformations

L: P2 -> P3 defined by L((p(t)) = t³p'(0) + t²p(0)
(p'(t) means the derivative of p(t) with respect to t)

How is this a linear transformation? I tried doing L[p(t) + q(t)] but it got so complicated since I got L[(t³p'(0) + t²p(0)) + (t³q'(0) + q²p(0))]..

Answer 1

what is P2 and P3?

Answer 2

Sorry, it means polynomials

Answer 3

hmm... let me see.
Since the original polinomial is P2: for example p(t)=at^2 +bt +c
the p'(t)=2at+b
so L(p(t)) = t^3b + t^2c
now: let q(t) = dt^2+et+f
L(p(t) +q(t)) = L((a+d)t^2+(b+e)t +(b+c))
can you continue?

Answer 4

Oh, I think I got it!

Answer 5

litle correction in L(p(t) +q(t)) = L((a+d)t^2+(b+e)t +(b+c))
should be L(p(t) +q(t)) = L((a+d)t^2+(b+e)t +(c+f))
then: L((a+d)t^2+(b+e)t +(c+f)) = t^3(b+e)+t^2(c+f)=t^3b +t^3e + t^2c+t^2f=
=(t^3b+t^2c) + (t^3e+t^2f) = L(p(t)) +L(q(t))

Answer 6

Oh alright, thanks so much myko!

Answer 7

yw