Exercise 3.E.1

Proof. Suppose $T$ is linear. Let $(v_1,T{v}_1),(v_2,T{v}_2)\in \mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$. We have

where the second line follows because $T$ is linear and the third by definition of $\mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$. Hence $\mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$ is closed under addition. Similarly, it also closed under scalar multiplication. Therefore, $\mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$ is a subspace of $V\times W$.

Conversely, suppose $\mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$ is a subspace of $V\times W$. Let $v_1,v_2\in V$. Then

Since $\mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$ is a subspace of $V\times W$, adding the first two vectors above shows that

Because $T$ is function, if $(v,w),(v,ŵ)\in \mathrm{graphof}<!--\; FUNCTION\; APPLICATION\; -->T$ then $w=ŵ$. This, together with $(1)$ and $(2)$, implies that $T(v_1+v_2)=T{v}_1+T{v}_2$. Hence $T$ satisfies the additivity property of linear maps. Similarly, $T$ satisfies the homogeneity property. Therefore, $T$ is indeed a linear map. □