Orthogonal Complement Properties: Explanation

Let's analyze what the properties of the orthogonal complement operation mean.

The property ‘If \(S\subseteq T\subseteq \VSpc{W}\), then \(\OrthCmpl{T}\subseteq \OrthCmpl{S}\)’. We read this as:

If S is a subset of T or equal to T, and T is a subset of W or equal to W, then the orthogonal complement of T is a subset of the orthogonal complement of S or is equal to the orthogonal complement of S.

Let’s think this through:

1. \(\OrthCmpl{T}\) consists of all those vectors of \(\VSpc{W}\) which are orthogonal to every vector in \(\VSpc{T}\)
2. If every vector in \(S\) is also in \(T\), this means that a vector in the orthogonal complement of \(T\) is also in the orthogonal complement of \(S\).

The property \(\OrthCmpl{S} = \OrthCmpl{\SpanOf{S}}\).   We read this as:

The orthogonal complement of S is equal to the orthogonal complement of the span of S.

We know that \(S\) is contained in \(\SpanOf{S}\). So we see that \(\OrthCmpl{\SpanOf{S}}\) is contained in \(\OrthCmpl{S}\) from the previous property. That \(\OrthCmpl{S}\) is also contained in \(\OrthCmpl{\SpanOf{S}}\) requires proof by computation.