Define the following: \[ T:p_3 \rightarrow p_2\\by\\T(a+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \]
Find the pre-image of 0. Does this linear transformation seem familiar?
Let the following be the preimage of T \[ T^{-1}(0) \]
The preimage of T is the set of polynomials where the following is true \[ T(ax+bx+cx^{2}+dx^{3})=0 \]
This implies that: \[ b+2cx+3dx^{2}=0 \] Therefore the set of all polynomials where a = 0,b = 0,and c = 0 is the polynomials of degree 0.
Does this transformation look familiar? \[ T(a+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \] This is similar to: \[ dx/dy(a+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \]
The expression on the right is the derivative of the polynomial on the left assuming a, b, and c are non zero constants as the derivative of a constant is zero.
\[
dx/dy(ax^{n})=nax^{n-1}
\]