Supposing that $T$ is the identity on $N$ and on $V/N$, and setting $R=I-T$, show that there exists a $K\in Hom(V/N, V)$ such that $R=K\circ\pi$. Show that for any coset $A$ of $N$ the action of $T$ on $A$ can be viewed as translation through $K(A)$. that is, if $\xi\in A$ and $\eta = K(A)$, then $T(\xi)=\xi+\eta$
This question is taken from advanced calculus by Lynn Harold Loomis and Shlomo Sternberg
The book designate $\overline{\alpha}=\alpha+N$ and $\pi(\alpha)=\overline{\alpha}$ (V is assumed to be a vector space and $N$ is a subspace of $V$)
For the first part, I have shown that $K(\overline{\alpha})=R(\alpha)$ is well defined.
The second part:
We know that $R=I-T\Rightarrow T=I-R=I-K\circ\pi$
Therefore $T(\xi)=\xi-K(\overline{\xi})=\xi-K(A)=\xi-\eta\neq\xi+\eta$
Is this a mistake in the book or am I missing something?