Results 1 to 3 of 3

Thread: Can someone solve this problem for me?

  1. #1
    Newbie
    Joined
    Apr 2018
    From
    sydney
    Posts
    1

    Can someone solve this problem for me?

    1. Let VV be the three-dimensional space of real-valued polynomials of a real variable of degree 2≤2. Let T:VVT:V→V be the translation operator that acts on a polynomial ff as (Tf)(t)(Tf)(t) = f(t+1)f(t+1). (a) Construct the matrices [T]V[T]V of TT with respect to the basis V=1,t,t2V=1,t,t2.

    (b) Show that this matrix has only one eigenvalue (the multiplicity of which is 3). Show that the dimension of the corresponding eigenspace is 11 and find a non-zero vector belonging to this space.

    (c) Use the above to define the eigenvalue and the eigenspace of TT


    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    11,119
    Thanks
    718
    Awards
    1

    Re: Can someone solve this problem for me?

    We generally don't give full solutions unless there is a good reason to. Instead, please let us know what you've been able to do with this. That will help us help you better.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    19,714
    Thanks
    3002

    Re: Can someone solve this problem for me?

    A polynomial of degree less than or equal to 2 is of the form $\displaystyle ax^2+ bx+ c$ which you can represent as $\displaystyle \begin{bmatrix} a \\ b \\ c \end{bmatrix}$. A linear operator on that can be represented as a matrix, $\displaystyle \begin{bmatrix} p & q & r \\ s & t & u \\ v & w & x \end{bmatrix}$.

    You are told that this operator maps $\displaystyle ax^2+ bx+ c$ to $\displaystyle a(x+ 1)^2+ b(x+ 1)+ c= ax^2+ 2ax+ a+ bx+ b+ c= ax^2+ (2a+ b)x+ a+ b+ c$ which would be represented by \begin{bmatrix}a \\ 2a+ b\\ a+ b+ c \end{bmatrix}

    So you want to find numbers, p, q, r, s, t, u, v, w, x such that
    $\displaystyle \begin{bmatrix} p & q & r \\ s & t & u \\ v & w & x \end{bmatrix}\begin{bmatrix}a \\ b \\ c \end{bmatrix}= \begin{bmatrix}a \\ 2a+ b\\ a+ b+ c\end{bmatrix}$
    for all a, b, and c. I suggest using a= 1, b= c= 0, b= 1, a= c= 0, and c= 1, a= b= 0, in turn.
    Last edited by HallsofIvy; Apr 12th 2018 at 04:22 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. I don't know how to solve this problem
    Posted in the Business Math Forum
    Replies: 1
    Last Post: Mar 13th 2017, 09:29 PM
  2. How do i solve the following problem?
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: Nov 15th 2015, 09:40 PM
  3. problem i need to solve
    Posted in the Algebra Forum
    Replies: 2
    Last Post: Mar 20th 2009, 03:54 AM
  4. Some help with a problem I'm trying to solve?
    Posted in the Geometry Forum
    Replies: 0
    Last Post: Apr 10th 2008, 06:32 PM
  5. Some help with a problem I'm trying to solve?
    Posted in the Geometry Forum
    Replies: 0
    Last Post: Apr 10th 2008, 06:29 PM

/mathhelpforum @mathhelpforum