# Thread: Can someone solve this problem for me?

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

2. ## 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

3. ## 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.