definition of projection of vector into vectors subspace w.r.t. complementary subspace
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Note
Starting Context
- The reader knows a definition of complementary subspace of vectors subspace.
Target Context
- The reader will have a definition of projection of vector into vectors subspace with respect to complementary subspace
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V'\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( V\): \(\in \{\text{ the vectors subspaces of } V'\}\)
\( \widetilde{V}\): \(\in \{\text{ the complementary subspaces of } V\}\)
\( v'\): \(= v + \widetilde{v}\), \(\in V'\)
\(*v\):
//
Conditions:
\(v \in V \land \widetilde{v} \in \widetilde{V}\)
//
2: Natural Language Description
For any field, \(F\), any \(F\) vectors space, \(V'\), any vectors subspace of \(V'\), \(V\), any complementary subspace of \(V\), \(\widetilde{V}\), and any element, \(v' = v + \widetilde{v} \in V'\) such that \(v \in V\) and \(\widetilde{v} \in \widetilde{V}\), \(v\)
3: Note
Once \(V\) and \(\widetilde{V}\) are specified, the projection is uniquely determined, as is described in Note for the definition of complementary subspace of vectors subspace.
A complementary subspace is need to be specified in order to talk about projection, because the projection depends on the choice of complementary subspace.
For example, when \(V'\) is the span of the basis, \((e_1, e_2)\), and \(V\) is the span of the basis, \((e_1)\), if \(\widetilde{V}\) as the span of the basis, \((e_2)\), is the complementary subspace, the projection of \(e_1 + e_2\) is \(e_1\), but if \(\widetilde{V}\) as the span of the basis, \((e_1 + e_2)\), is the complementary subspace, the projection of \(e_1 + e_2\) is \(0\).
