Polynomial Mapping: Linear Algebra Basics for Beginners

[zairizain]

Hi

I'm a new student and don't have any basics for linear algebra. thanks

Homework Statement

Q1. Let X be the vector space of polynomials of order less than or equal to M.

a) show that the set B = {1,x,...x^M}

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

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

(i) Show that T is linear
(ii) Derive 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

Thanks again

Homework Equations

The Attempt at a Solution

 

[lippyka]

a) The set B = {1,x,...x^M} is a basis for the vector space X. This can be shown by noting that the set B is linearly independent and spans the vector space X. For linear independence, we must show that if a1*1 + a2*x +...+am*x^m = 0, then a1 = a2 = ... = am = 0. By expanding this equation, it is clear that each coefficient must be zero, which implies linear independence. For spanning, we must show that every polynomial of order less than or equal to M can be expressed as a linear combination of the elements of B. This is true since any polynomial can be written as a sum of powers of x up to x^M, and each power of x is present in B. Therefore, B is a basis for X.b) (i) To show that T is linear, we must show that T(a1*f1 + a2*f2) = a1*T(f1) + a2*T(f2). We can do this by writing out the definitions of T and f1 and f2 and then expanding the equation. Since the derivative of a sum is the sum of the derivatives, we get the desired result. Therefore, T is linear.(ii) To derive the matrix representation for T, we must first write out T in terms of the basis B. Since T is the derivative operator, we can write it as T = d/dx. Then, for any polynomial g(x) = a0 + a1*x +...+am*x^m, we have Tg(x) = a1 + 2*a2*x +...+ m*am*x^(m-1). Now, we can write this in terms of the basis B. We get Tg(x) = a1*x + 2*a2*x^2 +...+ m*am*x^m. Finally, we can write this as a matrix equation. Letting A = [a1, a2,...am], we get Tg(x) = [x, 2*x^