SOLUTION: Determine whether the following functions T : V -> W define linear transformations (be sure to justify your answer fully): V = W = P(Rational), T(f)(x) = (x + 1)f(x) −

Algebra ->  College  -> Linear Algebra -> SOLUTION: Determine whether the following functions T : V -> W define linear transformations (be sure to justify your answer fully): V = W = P(Rational), T(f)(x) = (x + 1)f(x) −       Log On


   

Question 27388: Determine whether the following functions T : V -> W define linear transformations
(be sure to justify your answer fully):

V = W = P(Rational), T(f)(x) = (x + 1)f(x) − f(1)x^2.
Thanks a lot to who ever is solving this. Means a great deal.
Thank you.
Answer by khwang(438) About Me  (Show Source):
You can put this solution on YOUR website!
If f, g be two polynomials in Q, then
T(f+g)(x) = (x + 1)(f+g)(x) − %28f%2Bg%29%281%29x%5E2
= (x + 1)(f(x)+g(x)) − %28f%281%29%2Bg%281%29%29x%5E2+
= (x+1) f(x) - f%281%29+x%5E2++ (x+1)g(x) - g%281%29+x%5E2
= T(f)(x) + T(g)(x)
= (T(f)+T(g))(x),
Hence, T(f+g) = T(f)+T(g).
and for a in Q,
T(af)(x) = (x + 1)(af)(x) - %28af%29%281%29x%5E2
= a(x+1) f(x) - af%281%29x%5E2
= a((x+1)f(x) - f%281%29x%5E2)
= a T(f(x))
Hence, T(af) = aT(f).
This shows T is linear from Q[x]=V to Q[x].
(Or check T(af+bg)(x) = a T(f)(x) + b Tg(x) directly.
Kenny

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