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**The problem statement, all variables and given/known data**

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 -1

What 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)?