How can I understand linear functions and their matrices in polynomials?

[mrroboto]

I should just give math up, I don't understand this at all. It seems like for a) the function could be squared, and other than that it doesn't make any sense.


Let V = {p element of R[x] | deg(p) <=3} be the vector space of all polynomials of degree 3 or less.

a) Explain why the derivative d/dx: V -> V is a linear function

b) Give the matrix for d2/dx2 in the basis {1,x,x^2, x^3} for V

c) Give the matrix for the third derivative d2/dx2: V->V using the same basis

d) Give the matrix for the third derivative d3/dx3: V->V using the same basis

e) Give a basis for ker(d2/dx2)

f) what is the matrix for the linear map (d/dx + 4(d3/dx)): V->V

g) Let T: R[x] ->R[x] be the function T(p(x))= integral from -2x to 2 of p(t)dt

Explain why T is an element of L(R[x])

 

[CompuChip]

OK, let's start with a)
What does it mean for a function to be linear?
What is the formula you have to check, that needs to be true for a linear function f?

 

[mrroboto]

That is is closed under vector addition and scalar multiplication? In this case, the derivative's dimension will always be a square (or less), and it is closed under addition because added two squares will always be within a cube, and it is closed under multiplication because any constant times a square will still be within a cube.

 

[HallsofIvy]

I think your problem is that you don't learn the definitions well enough! In mathematics, all definitions are "working" definitions- you use the precise words of the definition in proofs and problems.

My point is that you were asked, "What does it mean for a function to be linear?" by Compuchip and answered, "That is is closed under vector addition and scalar multiplication? " That has nothing to do with functions at all- that is a definition of "subspace". A function f(x) on a vector space is linear if and only if f(u+ v)= f(u)+ f(v) and f(av)= af(v) for any vectors u and v and scalar a. If p(x) and q(x) are in your set of polynomials, what is d(p+q)/dx? If, also, a is a number, what is d(ap(x))/dx?

A standard way of finding the matrix representing a linear transformation in a given basis is to apply the linear transformation to each basis vector in turn. The coefficients of the result, written in that basis, form the colums of the matrix.