Suppose $T$ is a function from $V$ to $U$. The graph of $T$ is the subset of $V \times W$ defined by

Prove that $T$ is a linear map if and only if the graph of $T$ is a subspace of $V \times W$.

Suppose $T$ is a linear map, then the graph of $T$ is a subspace of $V \times W$ because $(v_1,Tv_1) + (v_2,Tv_2) = (v_1+v_2,T(v_1+v_2))$ is in the graph, likewise for scalar multiplication.

Now suppose the graph is a subspace of $V \times W$, because the graph is closed under addition we have (for all $v_1,v_2 \in V$)

Since the graph of $T$ is defined as all pairs $(v,Tv)$ we must have $T(v_1+v_2)=Tv_1+Tv_2$ for all $v_1,v_2$. Thus $T$ is additive, a similar argument works for scalar multiplication. Thus $T$ is a linear map