Linear algebra - squaring via transformations

[Niles]

Homework Statement

I have a transformation (not linear! that is what I have to show) F given by:

F : P_4 -> P_7 (P_7 is the vector-space spanned by polynomials less than degree 7). I also know that F(p(x)) = (p(x))^2.

The matrix A representing F with respect to the two basis is the one I get by taking the transformation F on P_4's elements [x^3, x^2, x, 1] and expressing by P_7's elements. I get a 7x4-matrix with 4 zeroes and the rest are zero-entries.

The Attempt at a Solution

This matrix is the matrix A in L(x) = Ax. So if I take a polynomial in P_4 and multiply with A, it should be squared:

A*(a_1*x^3, a_2*x^2, a_3*x, a_0)^T.

But this doesn't make a_1*x^3 go to (a_1)^2*x^6 and so on? Where am I going wrong?

I hope you understand my questions.


Niles.

 

[morphism]

I don't really understand what you're asking. You're saying that F is not linear, but then you say it's represented by the matrix A.

What are you trying to do exactly? Prove that the mapping F:P_4->P_7 (where P_n is the space of polynomials over R of degree <n) given by F(p(x))=(p(x))^2 is linear? Well, it's not! For example: F(x+x) = F(2x) = 4x^2, which is not equal to F(x)+F(x)=2x^2.