i. If \(S\subseteq T\) then \(\OrthCmpl{T}\subseteq \OrthCmpl{S}\)
If \(\Vect{x}\in \OrthCmpl{T}\), we need to show that \(\Vect{x}\in \OrthCmpl{S}\). So we need to show that \(\DotPr{ \Vect{x} }{ \Vect{s} } = 0\), for every \(s\in S\). – But if \(\Vect{s}\in S\), then also \(\Vect{s}\in T\), and so \(\DotPr{ \Vect{x} }{ \Vect{s} } = 0\) because \(\Vect{x} \in \OrthCmpl{T}\).
ii.
We need to verify the two properties below
- \(\OrthCmpl{S}\) contains \(\OrthCmpl{\SpanOf{S}}\)
- \(\OrthCmpl{\SpanOf{S}}\) contains \(\OrthCmpl{S}\)
The inclusion 1. holds by part (i), using that \(S\subseteq \SpanOf{S}\). To see that inclusion 2. holds, consider \(\Vect{x}\in \OrthCmpl{S}\). We need to show that \(\DotPr{ \Vect{x} }{ \Vect{v} } = 0\) for every \(\Vect{v} \in \span(S)\). Now \(\Vect{v} = t_1 \Vect{s}_1 + \cdots + t_k \Vect{s}_k\), for some \(\Vect{s}_1\), ... , \(\Vect{s}_k\) in \(S\), and \(t_1\), ... , \(t_k\) in \(\RNr\). Therefore
\(\DotPr{ \Vect{x} }{ \Vect{v} }\) | \(=\) | \(\DotPr{ \Vect{x} }{t_1 \Vect{s}_1 + \cdots + t_k \Vect{s}_k) }\) |
\(\) | \(=\) | \(t_1(\DotPr{ \Vect{x} }{ \Vect{s}_1 }) + \cdots + t_k(\DotPr{ \Vect{x} }{ \Vect{s}_k })\) |
\(\) | \(=\) | \(0\) |
This proves part (ii) of the proposition.
iii. \(\VSpc{V}\cap \OrthCmpl{\VSpc{V}} = \Vect{0}\).
Suppose \(\Vect{x}\) belongs to \(\VSpc{V}\) and to \(\OrthCmpl{\VSpc{V}}\), then \(\DotPr{ \Vect{x} }{ \Vect{x} } = 0\), and
so
\(\Vect{x} = \Vect{0}\).
iv. \(S^{\bot} = S^{\bot\bot\bot}\)
We need to show that the two inclusions below hold
- \(\OrthCmpl{S}\subseteq S^{\bot\bot\bot}\)
- \(S^{\bot\bot\bot}\subseteq \OrthCmpl{S}\)
Consider \(\Vect{x}\in \OrthCmpl{S}\). We need to show that \(\DotPr{ \Vect{x} }{ \Vect{y} }=0\) for all \(\Vect{y}\in S^{\bot\bot}\). But \(\Vect{y}\) is in \(S^{\bot\bot}\) because \(\DotPr{ \Vect{x} }{ \Vect{y} }=0\) for each \(\Vect{x}\in \OrthCmpl{S}\), and this is exactly what we wanted to show.
The converse inclusion \(S^{\bot\bot\bot}\subseteq \OrthCmpl{S}\) holds because \(S\subseteq S^{\bot\bot}\).
This completes the proof of the proposition.