definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its covectors space w.r.t. original space basis
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of dual basis for covectors (dual) space of basis for finite-dimensional vectors space.
- The reader knows a definition of %category name% isomorphism.
Target Context
- The reader will have a definition of canonical 'vectors spaces - linear morphisms' isomorphism between finite-dimensional vectors space and its covectors space with respect to original space basis.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( V\): \(\in \{\text{ the finite-dimensional } F \text{ vectors spaces }\}\)
\( V^*\): \(= L (V: F)\)
\( J\): \(\in \{\text{ the finite index sets }\}\)
\( B\): \(\in \{\text{ the bases for } V\}\), \(= \{b_j \vert j \in J\}\)
\( B^*\): \(= \text{ the dual basis of } B\), \(= \{b^j \vert j \in J\}\)
\(*f\): \(: V \to V^*, v^j b_j \mapsto \sum_{j \in J} v^j b^j\), \(\in \{\text{ the 'vectors spaces - linear morphisms' isomorphisms }\}\)
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Conditions:
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\(f\) depends on the choice of \(B\) (as will be seen in Note), which is the reason why the concept is called with "with respect to original space basis".
2: Note
\(f\) is indeed a 'vectors spaces - linear morphisms' isomorphism, by the proposition that between any vectors spaces, any map that maps any basis onto any basis bijectively and expands the mapping linearly is a 'vectors spaces - linear morphisms' isomorphism.
Let us see that \(f\) depends on the choice of \(B\).
Let \(B' = \{b'_j \vert j \in J\}\) be another basis for \(V\).
\(b'_j = b_l M^l_j\) for an invertible matrix, \(M\). So, \(b_j = b'_l {M^{-1}}^l_j\).
The dual basis of \(B'\), \(B'^* = \{b'^j \vert j \in J\}\), is \(\{{M^{-1}}^j_l b^l\}\), the proposition that for any finite-dimensional vectors space, the transition of the dual bases for the covectors space with respect to any bases for the vectors space is this.
The canonical 'vectors spaces - linear morphisms' isomorphism with respect to \(B'\), \(f': V \to V^*\), maps \(b_j = {M^{-1}}^l_j b'_l\) to \(\sum_{l \in J} {M^{-1}}^l_j b'^l = \sum_{l \in J} {M^{-1}}^l_j {M^{-1}}^l_m b^m\), which does not equal \(b^j\) in general. In fact, when \(M\) is orthogonal, \(M^m_l = {M^{-1}}^l_m\), and \(= \sum_{m \in J} {M^{-1}}^l_j M^m_l b^m = \sum_{m \in J} \delta^m_j b^m = b^j\).
