(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let X be the vector space of polynomial of order less than or equal to M

a) Show that the set B={1,x,...,x^M} is a basis vector

b) Consider the mapping T from X to X defined as:

f(x)= Tg(x) = d/dx g(x)

i) Show T is linear

ii) derive a matrix representation for T in terms of the basis B

iii) what are the eigenvalues of T

iv) compute one eigenvector associated with one of the eigenvalues

2. Relevant equations

3. The attempt at a solution

a) i)Linear independence;

a1(1) + a2(x)+...+an(x^M) = 0

a1=a2=an=0

ii)Span

a+bx+...+cx^M=0

Such that; a1(1) +a2(x)+...+an(x^M) = a+bx+...+cx^M

a1=a, a2=b, an=c

b)

i) f(x) = a0 + a1X+...+amX^M

g(x) = b0 + b1X+...bmX^M

g(t) = b0+b1t+....+bmt^M

Tg(t) = b0t + b1t^2+...+bmt^(M+1)

For any scalar, k is element K

T(k g(t)) = t (k g(t))

= k (t g(t))

= KT (g (t))

Thus T is linear

ii) B= {1, x ,x^2,...,x^M}

matrix T,=

0 0 0 ...0

0 1 0 ...0

0 0 2 ...0

0 0 0 ...0

. . . .. .

0 0 0 .. M

iii) eigenvalues of T, lambda = [0, 1, 2....M]

iv)for lambda = 1;

(A-lambda I)=0

(A- I ) = 0

[matrix T] [ a1;a2;...am] = [ a1;a2;...am]

a1=a2=...=am

eigenvector for lambda =1 is;

[1, 1, .....1]

Is this correct?Please help me.

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# Homework Help: Polynomial Basis and Linear Transformation

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