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.Com
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) (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) −
= (x + 1)(f(x)+g(x)) −
= (x+1) f(x) - + (x+1)g(x) -
= 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) -
= a(x+1) f(x) -
= a((x+1)f(x) - )
= 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
=
RELATED QUESTIONS
determine whether the following functions are linear transformations. justify your... (answered by 9003105888080)
T = 7000 (w / d^2)(1 / v - 1 / u)
Determine v.
Not sure how to solve.
(answered by mananth,Alan3354)
use slope to determine whether the given vertices form a parallelogram t(-1,0) , u(0,5) , (answered by solver91311)
Hi all, Im having trouble understanding what this question is asking, can anyone show me... (answered by Alan3354)
Determine whether the given point is a solution of the linear inequality. Make sure to... (answered by stanbon)
Let T, U: V to W be linear transformations. Prove that:
a. R(T+U) is a subset of... (answered by khwang)
determine whether the following set is a function and justify your answer
{(2,5),... (answered by longjonsilver)
Using the following formula:
T = 7000 w/d^2 (1/v - 1/u)
A projectile of weight w... (answered by ikleyn,MathTherapy)
determine whether the argument is valid or invalid. You may compare the argument to a... (answered by jim_thompson5910)