The projection on $U$ along $W$ is the function $T:V\rightarrow V$ defined by $T(u+w)=u$, where $u \in U$, $w \in W$. Let $V=\mathbb{R}^2$, and $U=\{(x,-x): x \in \mathbb{R}\}$, and $W=\{(x,0): x\in \mathbb{R}\}$. Prove that $V=U \oplus W$, and give formulas for $T$, the projection on $U$ along $W$, and $S$, the projection on $W$ along $U$.
Is it enough to show that T is linear to show that $V=U \oplus W$? I'm uncertain as to whether that is enough to show what I want. Thanks in advance.