definition of orthogonal complement of subset of vectors space with inner product
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of inner product on real or complex vectors space.
Target Context
- The reader will have a definition of orthogonal complement of subset of vectors space with inner product.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\mathbb{R}, \mathbb{C}\}\), with the canonical field structure
\( V\): \(\in \{\text{ the } F \text{ vectors spaces }\}\), with any inner product, \(\langle \bullet, \bullet \rangle\)
\( S\): \(\subseteq V\)
\(*S^\perp\): \(= \{v \in V \vert \forall s \in S (\langle v, s \rangle = 0)\}\)
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Conditions:
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2: Note
\(S\) does not need to be any vectors subspace.
But when \(S \neq \emptyset\), \(S^\perp = (S)^\perp\), where \((S) = Span (S)\) (see Note for the sub-'vectors space' generated by \(S\)).
That is because for each \(v \in S^\perp\), for each \(s \in (S)\), \(\langle v, s \rangle = 0\), because \(s = r^1 s_1 + ... + r^n s_n\) where \(s_j \in S\), and \(\langle v, s \rangle = \langle v, r^1 s_1 + ... + r^n s_n \rangle = r^1 \langle v, s_1 \rangle + ... + r^n \langle v, s_n \rangle = 0 + ... + 0 = 0\); for each \(v \in (S)^\perp\), for each \(s \in S\), \(\langle v, s \rangle = 0\), because \(s \in (S)\).
