You aren't trying to find the kernel of an arbitrary matrix! (per se.)
Right, you have this linear map, denoted L, and you have said what it does. You now wish to find its kernel. That is, which matrices M are such that L(M)=0? That is, for which values of $\displaystyle a, b, c, d$ are such that $\displaystyle ax^3+bx^2+cx+d = 0$?
(I now have to go, sorry!)
With all due respect I think the OP is way more lost in this that what we could have guessed at the beginning, and I think he doesn't need help in this question but rather a thorough check over the whole material of this course, perhaps from the start...and the sooner the better, as things are going to get tougher pretty quick and without the elementary foundations is almost impossible to succeed.
I don't believe that a "thorough review" of all this stuff can be given within the frame of MHF but rather, imo, with the help of some private tutoring.
Tonio
Here is a basic outline of what you should be doing:
To prove two vector spaces $\displaystyle U$ and $\displaystyle V$ are isomorphic you must
(1) Find a linear map between them, $\displaystyle T: U \rightarrow V$,
(2) Prove that $\displaystyle T$ is an injection,
(3) Prove that $\displaystyle T$ is a surjection.
In some ways, step (1) is the hardest - it requires some imagination. For the question you are looking at, you have done this;
$\displaystyle T:\left( \begin{array}{cc}
a & b\\
c & d\end{array} \right) \mapsto ax^3+bx^2+cx+d$
So this is your linear map. You must, however, prove that it is linear. That is, you have to prove that
(1.a) $\displaystyle T(u_1) + T(u_2) = T(u_1+u_2)$,
(1.b) $\displaystyle \lambda T(u) = T(\lambda u)$.
For (2), $\displaystyle T$ is an injection if and only if $\displaystyle \left(T(u_1) = T(u_2) \Rightarrow u_1=u_2 \right)$, by definition. Now, as your map is a linear map, $\displaystyle T(u_1) = T(u_2) \Leftrightarrow T(u_1) - T(u_2) = 0 \Leftrightarrow T(u_1-u_2) = 0$. Therefore, $\displaystyle T$ is an injection if and only if the set $\displaystyle K=\{u: T(u) = 0\}$ consists only of the zero of the linear space $\displaystyle U$ (as then $\displaystyle T(u_1) = T(u_2) \Rightarrow T(u_1-u_2) = 0 \Rightarrow u_1-u_2=0 \Rightarrow u_1=u_2)$). This set $\displaystyle K$ is called the kernel of your linear map, $\displaystyle T$.
Thus, what you need to prove is that if $\displaystyle T(u) = 0 = 0x^3+0x^2+0x+0$ then $\displaystyle u=0 = \left( \begin{array}{cc}
0& 0\\
0& 0\end{array} \right)$.
Does that make sense?
Now, for (3) it is sufficient to prove that each of the generators of the space $\displaystyle V$ is mapped onto by some element of $\displaystyle U$ (why?). So, what are the generators of $\displaystyle V$ in this case, and which elements of $\displaystyle U$ are mapped to them?
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I would agree with Tonio, you do need proper help. We can, and are willing, to help you here. However, you need to give more constructive replies. `I'm confused' doesn't really count as constructive. You have to say why you are confused.