Representing map F in another basis

[simpledude]

Homework Statement
Let V = P(3)(R) be the vector space of all polynomials P : R −> R with degree less than
3. We consider the mapping F : V −> V defined for all P belonging to V by F(P(x)) = P(−x) for all x 2 R.

I have to represent the linear operator F as a matrix in the basis {1 + x, 2x, x2, 1 + x3}

The attempt at a solution

I think I got this one right but would like your professional input.
So I apply F to the 4 vectors.

F(1+x) = 1-x
F(2x) = -2x
F(x^2) = x^2
F(1+x^3) = x^3

This gives a matrix of

1 0 0 1
-1 -2 0 0
0 0 1 0
0 0 0 -1What I am confused about, is the fact that it says only polynomials of degree less
than 3, so does that mean I do not consider F(1+x^3) ? i.e. I would only have a 3 x 3 matrix (no last column and last row)?

 

[mighty2000]

Your approach to representing the linear operator F as a matrix is correct. However, you are correct in questioning whether or not to include F(1+x^3) in the matrix. Since the vector space V is defined as polynomials with degree less than 3, it would be more accurate to only include the first three vectors in the matrix. This would give a 3x3 matrix as you mentioned. Including F(1+x^3) would not change the result of the matrix, but it would be more consistent with the definition of V to only include polynomials of degree less than 3.