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Matrix (Complex Numbers)

  1. Nov 28, 2007 #1
    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 doesnt 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])
     
  2. jcsd
  3. Nov 28, 2007 #2

    CompuChip

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    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?
     
  4. Nov 28, 2007 #3
    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.
     
  5. Nov 28, 2007 #4

    HallsofIvy

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    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.
     
    Last edited: Nov 28, 2007
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