Exercise LT.C43
Define the following: \[ T:p_3 \rightarrow p_2\\by\\ T(ax+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \]
Find the pre-image of 0. Does this linear transformation seem familiar?
Lets understand what the pre-image is http://mathworld.wolfram.com/Pre-Image.html
A Pre-image exists if whether f has an inverse or not. If f does not have an inverse, then the preimage is defined as the exact image of y.
Let the following be the preimage of T \[ T^{-1}(0) \]
The preimage of T is the set of all polynomials where the following holds true \[ T(ax+bx+cx^{2}+dx^{3})=0 \]
This implies that the following is also true \[ b+2cx+3dx^{2}=0 \] Therefore it can be said that zero represents the 0 polynomial, in otherwords the set of all polynomials where a=0,b=0,and c=0 thus polynomials of degree 0.
How does this transformation look familiar? \[ T(ax+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \] Is the same as \[ dx/dy(ax+bx+cx^{2}+dx^{3})=b+2cx+3dx^{2} \]
In other words, the expression on the right is the derivative of the polynomial on the left assuming a, b, and c are non zero constants. The derivative of a constant is zero. The derivative of the other terms can be found using the power rule. \[ dx/dy(ax^{n})=nax^{n-1} \]