## Transcribed Text

Let Pn denote the vector space of polynomials of degree < n in the variable x. Let B and C
be the bases for P3 and P2 given by
B = {x³ - 3x, 3x2 + x, x2 - x, 1 + 2x}
C = {x2 - 1,1,2x}
Find the matrix of the linear transformation d P3 P2, f(x) f' (x) with respect to the
bases B and C
Theorem 1. Any nonzero polynomial f(x) = ao + aix +
anx of degree at most n has
at most n roots. In other words, if there are n + 1 distinct numbers bo, b1, bn and f(bo) =
f(b1) =
= f (bn) = 0 then f (x) is the zero polynomial.
Assume xo, T1, In are given distinct real numbers. Define the operator T: Pn Rn+1 by
p(xo)
p(x1)
T(p(x)) =
p(xn)
It can be shown (you don't have to do this) that T is a linear operator (for example: problem
10 in section 5.4 does this with xo = - -3,x1 = - 1, x2 = 1, x3 = 3 and n = 3.)
(a) Show the operator T is one-to-one (injective).
(b) Conclude that T is also onto (surjective).
(c) Let us define the matrix
1 xo x2
n
xo
1 X1 x2 1
xn
n
1
A =
1 x2 x2
x2
n
:
:
1 In xn
x n n
Explain how A relates to T and show that A is invertible.
Hint: showing A is invertible is quite difficult without explaining how A and T are related:
I don't advise you to try it!

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