Demonstrate that t T: P3 -> P3, Tp = p' is a linear function.
I know that to demonstrate if something is a linear function I must, check if the sum and multiplication with a scalar is linear. But I am a bit concerned cause when I derivate P3 it will always give me a P2, and this does not make sense to me. Can someone help me?
To demonstrate that a function is linear, you need to verify two properties: preservation of addition and preservation of scalar multiplication.
1. Preservation of Addition:
Let p, q be two polynomials in P3, and let a, b be scalars. We need to show that T(p + q) = Tp + Tq.
First, let's compute T(p + q):
T(p + q) = (p + q)'
= p' + q'
Now, let's compute Tp + Tq:
Tp + Tq = p' + q'
Since T(p + q) = Tp + Tq, the preservation of addition property holds.
2. Preservation of Scalar Multiplication:
Let p be a polynomial in P3 and let k be a scalar. We need to show that T(kp) = kTp.
First, let's compute T(kp):
T(kp) = (kp)'
= k(p')
Now, let's compute kTp:
kTp = k(p')
Since T(kp) = kTp, the preservation of scalar multiplication property holds.
Since T satisfies both properties, we can conclude that T is a linear function.
Regarding your concern about the derivative always resulting in a polynomial of degree one less: In this case, we are only considering the derivative of the polynomial and not the polynomial itself. So, while p belongs to P3, its derivative p' belongs to P2 because when you differentiate a polynomial of degree n, the resulting polynomial has degree n-1.