Why is T not invertible?
The linear transformation T:M(2x2) --> P2 defined by
[c d] = a+2bx+(c+d)x^2
I have to prove that this is not invertible. I don't even know where to start because if I was trying to prove that it was invertible, I wouldn't be able to solve for the determinant since I don't know how I'm supposed to put the polynomial into a matrix since it seems to me that it would only have 3 entries. What should I do to show that this is not invertible?
I think the idea is that, while a and b are uniquely determined, given any ol' quadratic, there's no way to tell how much of the coefficient of x^2 should be c, and how much should be d. Example:
quadratic is x^2+x-6. Obviously, a = -6, b = 1/2. But with c and d, I could have c=1, d=0, or c=0, d=1, or any other of an infinite number of possibilities.
To put it on a more rigorous footing, you could always write the RHS in terms of a standard basis for P2.
Oh that makes sense. That would make it impossible to find an inverse of T.