Orthogonal Vector Decomposition - Explanation

The proposition on orthogonal vector decomposition says that given subspaces \(\VSpc{V}\subseteq \VSpc{W}\) of \(\RNrSpc{n}\), every \(\Vect{x}\) in \(W\) has a unique sum expression

\(\Vect{x}\)

\(=\)

\(\Vect{x}_V + \Vect{x}_{\bot}\)

where \(\Vect{x}_V=\OrthoPrjctn{V}{\Vect{x}}\) is in \(\VSpc{V}\) and \(\Vect{x}_{\bot}\DefEq \Vect{x}- \OrthoPrjctn{V}{\Vect{x}}\) is in \(\OrthCmpl{V}\).

Let us express in words what this proposition says: given an arbitrary \(\Vect{x}\) in \(W\), there are two uniquely determined vectors \(\Vect{x}_V\) and \(\Vect{x}_{\bot}\) satisfying the following two conditions

1. \(\Vect{x}_V\) is in \(V\) and is computed as the orthogonal projection \(\Vect{x}_V \DefEq \OrthoPrjctn{V}{\Vect{x}}\) of \(\Vect{x}\) onto \(\VSpc{V}\).
2. \(\Vect{x}_{\bot}\) belongs to the orthogonal complement of \(\VSpc{V}\) in \(\VSpc{W}\). It is computed as

   \(\Vect{x}_{\bot}\)

   \(=\)

   \(\Vect{x} - \OrthoPrjctn{V}{\Vect{x}}\)

As a consequence, \(\Vect{x} = \Vect{x}_V + \Vect{x}_{\bot}\) has been ‘split’ into the sum of \(\Vect{x}_V\) in \(\VSpc{V}\), and \(\Vect{x}_{\bot}\) in \(\OrthCmpl{V}\).