So I'm working on homework for my Introductory Linear Algebra Course. The text is Gareth Williams Linear Algebra with Application – 7th Edition. I'm currently working section 4.7: Kernel, Range and the Rank/Nullity Theorem.
Question: Problem 20.
Prove that $T: P_3\rightarrow P_2$ defined as followed in linear. Find the Kernel and the Range of T. Give bases for these subspaces.
$$
T(a_3x^3 + a_2x^2 + a_1x + a_0) = a_3x^2 – a_0
$$
I'm fairly comfortable with proving that the above is linear, that is I can show that the transformation preserves the operations of addition and scalar multiplication. I show this by taking 2 polynomials in $P_1$ and $P_2$ and showing that $T(P_1 + P_2)$ = $T(P_1) + T(P_2)$ and that $T(cP_1)$ (where $c$ is an element of the Reals) $= cT(P_1)$
Now I understand that the kernel is the subset that maps to 0, which I believe in our case should be the zero polynomial. I'm stuck as to how about finding the kernel. I know that the range shares the dimension of the range and that the dimension of the range is equal to the rank. What I don't know how to do is to deal with this using polynomials. I'm comfortable with vectors that I can use to build a transition matrix, I just don't know how to work this through with polynomials. I feel like I could comfortably find a basis again with vector or matrix spaces but transferring these ideas to polynomials and function I struggle with. Help?
Best Answer
The kernel of this map is the following: $$ker(T) = \{ a_0 + a_1 x + a_2 x^2 + a_3 x^3 \in P_3 : a_3 x^2 - a_0 = 0\}$$
If this polynomial $a_3 x^2 - a_0$ is equal to zero for all $x$ values, then we know that $a_3$ and $a_0$ must be zero. So the kernel is in fact the set of all polynomials in $P_3$ with $a_3 = a_0 = 0$, in other words, all polynomials of the form $a_1 x + a_2 x^2$.
You can somewhat check this result using rank nullity. The dimension of $P_3$ is equal to $4$. The dimension of the image is $2$, and hence the dimension of the kernel is $4-2 = 2$.