= Solution
The <kernel of a linear map> $g$ is precisely $U$, since
$$
g(u_1,u_2)=0\quad\Longleftrightarrow\quad u_2=-u_1\quad\Longleftrightarrow\quad (u_1,u_2)=(u,-u),\ u\in W_1\cap W_2.
$$
Part (c) gives $\operatorname{im}g=W_1+W_2$. Apply the <rank-nullity theorem> to the <finite-dimensional vector space> $W_1\times W_2$, and then parts (a) and (b):
$$
\dim W_1+\dim W_2=\dim(W_1\times W_2)=\dim\ker g+\dim\operatorname{im}g=\dim(W_1\cap W_2)+\dim(W_1+W_2).
$$
This proves the <dimension formula for a sum of subspaces>. The <kernel of a linear map> records exactly the freedom to transfer a shared <vector> from one summand to the other without changing their <sum>.
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