Definition 6.2.1. (Natural Projection).
Let \(V\) be a vector space over a field \(F\) and \(W\leq V\) be a subspace. The map
\begin{equation*}
\pi_W\colon V\to V/W
\end{equation*}
given by
\begin{equation*}
v\mapsto v+W
\end{equation*}
is called the natural projection.Suppose that \(\overline{U}\) is a subspace of \(V/W\text{.}\) Let \(X=\{u:u+W\in\overline{U}\}\text{,}\) i.e., \(X\) is the set of all coset representatives of \(\overline{U}\text{.}\) In particular, since \(0+W=w+W\) for any \(w\in W\text{,}\) we get that \(W\subset X\text{.}\) We claim that \(X\) is a subspace of \(V\text{.}\) Indeed, suppose that \(u_1,u_2\in X\text{,}\) i.e., \(u_1+W,u_2+W\in\overline{U}\text{.}\) Therefore, for any \(\alpha_1,\alpha_2\in F\text{,}\) we have \((\alpha_1u_1+\alpha_2u_2)+W\in\overline{U}\) and, by the definition of \(X\text{,}\) we get that \(\alpha_1u_1+\alpha_2u_2\in X\text{.}\) Hence, \(X\) is a subspace of \(V\) containing \(W\text{.}\)
Now suppose that \(X\) is a subspace of \(V\) containing \(W\text{.}\) Consider
\begin{equation*}
X/W=\{x+W:x\in X\}.
\end{equation*}
Verify that \(X/W\) is a subspace of \(V/W\text{.}\)
Checking other assertions is left to the reader.
