definition of complementary subspace of vectors 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 1
- 4: Note 2
- 5: Note 3
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of complementary subspace of vectors 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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V'\): \(\in \{\text{ the } F \text{ vectors spaces }\}\)
\( V\): \(\in \{\text{ the vectors subspaces of } V'\}\)
\(*\widetilde{V}\): \(\in \{\text{ the vectors subspaces of } V'\}\)
//
Conditions:
\(V \cap \widetilde{V} = \{0\}\)
\(\land\)
\(V' = V + \widetilde{V}\)
//
\(V' = V + \widetilde{V}\) means that \(\forall v' \in V' (\exists v \in V, \exists \widetilde{v} \in \widetilde{V} (v' = v + \widetilde{v}))\).
2: Natural Language Description
For any field, \(F\), any \(F\) vectors space, \(V'\), and any subspace, \(V \subseteq V'\), any subspace, \(\widetilde{V} \subseteq V'\), such that \(V \cap \widetilde{V} = \{0\}\) and \(V' = V + \widetilde{V}\)
3: Note 1
The decomposition, \(v' = v + \widetilde{v}\), is inevitably unique, because if \(v' = v_1 + \widetilde{v_1} = v_2 + \widetilde{v_2}\), \(v_1 - v_2 = \widetilde{v_2} - \widetilde{v_1}\), but the left hand side is an element of \(V\) and the right hand side is an element of \(\widetilde{V}\), so, it is in \(V \cap \widetilde{V} = \{0\}\), so, \(v_1 - v_2 = \widetilde{v_2} - \widetilde{v_1} = 0\), so, \(v_1 = v_2\) and \(\widetilde{v_1} = \widetilde{v_2}\).
4: Note 2
There can be multiple complementary subspaces for the same vectors subspace, because for example, \(V'\) is the span of the basis, \(\{e_1, e_2\}\), \(V\) is the span of the basis, \(\{e_1\}\), and the span of the basis, \(\{e_2\}\), is a complementary subspace of \(V\), but also the span of the basis, \(\{e_1 + e_2\}\), is a complementary subspace of \(V\).
5: Note 3
\(\widetilde{V}\) does not need to be "perpendicular" to \(V\) in order for it to be a complementary subspace of \(V\). In fact, "perpendicular" is defined only when inner product is defined on \(V'\), and inner product is not required for the concept of complementary subspace.
