Matrix Representation Question

[mau0706]

Homework Statement

This is a linear algebra matrix representation problem I have been trying to solve. I seem to keep getting it wrong, so I was hoping I could get some help.

The linear operator T: P₂-->P₂ is defined by T(P(x)) = xP'(x)-P(x). B={1,x,x²}, and B'={x,1+x,-1+x²} are two ordered bases for P₂.
Find the matrix representation for T relative to the ordered basis B'. ([T]_B')

The Attempt at a Solution

So far I have: xP'(x)-P(x) = ax²-c. Which as a matrix is: [0,-1,1;0,0,0;0,1,1] . I got this by putting ax²-c into [T(x)], [T(1+x)] and [T(-1+x²]. I have no idea if this is right or wrong, but I seem to keep making mistakes on this so any help is really appreciated.

I know how to get [T]_B which is [-1,0,0;0,0,0;0,0,1], I just keep messing up on [T]_B'.

 

[mighty2000]

Thank you for reaching out for help with your linear algebra problem. I understand that you have been struggling to find the correct matrix representation for the linear operator T with respect to the ordered basis B'. I would be happy to assist you in solving this problem.

Firstly, I want to clarify that your attempt at a solution is not entirely correct. The expression xP'(x)-P(x) is not a matrix, but a linear operator. To find the matrix representation of this operator, we need to apply it to the basis vectors of B' and express the resulting vectors in terms of the basis B'.

Let's start by finding the vector representation of T(x) with respect to B'. We know that T(x) = xP'(x)-P(x), so we can write it as a linear combination of the vectors in B':

T(x) = 0x + (-1)(1+x) + 1(-1+x2)

= -1 - x + x2

= (-1, -1, 1) with respect to the basis B.

Next, we can find the vector representation of T(1+x) with respect to B':

T(1+x) = (1+x)P'(1+x)-P(1+x)

= (1+x)(1) + (-1)(1+1+x) + 1(-1+1+x2)

= (2x-1, -1-x, 1+x2) with respect to the basis B.

Similarly, we can find the vector representation of T(-1+x2) with respect to B':

T(-1+x2) = (-1+x2)P'(-1+x2)-P(-1+x2)

= (-1+x2)(-1) + (-1)(1-1+x2) + 1(-1+x2)

= (-x2-1, -2, 2x) with respect to the basis B.

Now, we can arrange these vectors as columns to form the matrix representation of T with respect to B':

[T]B' = [(-1, 2x-1, -x2-1), (-1, -1-x, -2), (1, 1+x2, 2x)]

I hope this helps you in solving your problem. Please let me know if you have any further questions or if you need any clarification.[Your Name