So the question I am working on is:
Suppose $V = U_1\oplus U_2$. Define the linear operator $T$ on $V$ as follows: for every $v$ find $u_1\in U_1$ and $u_2\in U_2$ such that $v = u_1+u_2$, then $T(v)=u_1$.
Part (a) asks about the kernel of $T$. My professor taught me that the kernel of $T\colon V\to W$ (linear operator) is the set of vectors $v\in V$ such that $T(v)=0_W$.
When thinking of this question, I first thought about what the linear operator was doing to the input $v$. Because $v$ is the sum of $u_1$ and $u_2$, and the output is $u_1$, the linear operator was getting the sum of $u_1$ and $u_2$, and subtracting $u_2$ from it to get $u_1$. So I thought that the kernel of $T$ is $$ \{ u_1 + u_2 = v ~|\, u_1 = 0 \} $$
EDIT: $$ T(v) = u_1 $$ $$ T(u_1 + u_2) = T(u_1) + T (u_2) = u_1 $$ Would this mean that the $T(u_1)$ would be itself and the $T(u_2)$ would turn it to $0$?
You have to ask yourself: which are all the elements $v\in V$ such that $T(v)=0$? Clearly $Ker(V)=0\oplus U_2$, in fact let's take an element of this ensemble, $s=0+u_2$. By definition of the linear operator you have: $T(s)=T(0+u_2)=T(0)=0$