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Section 6.2 Natural Projection and Correspondence Theorem

Throughout this section we assume that \(V\) is a vector space over a field \(F\) and \(W\leq V\) a subspace.
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.
Remark 6.2.2.
It is easy to verify the following.

  1. \(\pi_W\) is an \(F\)-linear map

  2. \(\displaystyle \ker(\pi_W)=W\)

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.