definition of predual basis for finite-dimensional vectors space of basis for covectors space
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of predual basis for finite-dimensional vectors space of basis for covectors space.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( d\): \(\in \mathbb{N} \setminus \{0\}\)
\( V\): \(\in \{\text{ the } d \text{ -dimensional } F \text{ vectors spaces }\}\)
\( V^*\): \(= L (V: F)\)
\( J\): \(\in \{\text{ the } d \text{ -cardinality index sets }\}\)
\( B^*\): \(\in \{\text{ the bases for } V^*\}\), \(= \{b^j \vert j \in J\}\)
\(*B\): \(\in \{\text{ the bases for } V\}\), \(= \{b_j \vert j \in J\}\)
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Conditions:
\(\forall j \in J (\forall k \in J (b^j (b_k) = \delta^j_k))\)
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2: Note
While we have defined \(B^*\) from \(B\), by the definition of dual basis for covectors (dual) space of basis for finite-dimensional vectors space, this definition defines \(B\) from \(B^*\).
Let us see that \(B\) really exists uniquely for \(B^*\).
Let \(\{e_j \vert j \in J\}\) be any basis for \(V\): we already know that \(V\) and \(V^*\) have the same dimension.
Let \(\{e^j \vert j \in J\}\) be the dual basis of \(\{e_j \vert j \in J\}\) for \(V^*\).
\(b^j = M^j_l e^l\), where \(M\) is an invertible matrix over \(F\): invertible, because \(B^*\) is a basis for \(V^*\).
Let \(b_k = N^m_k e_m\), where \(N\) is a matrix over \(F\).
\(b^j (b_k) = M^j_l e^l (N^m_k e_m) = M^j_l N^m_k e^l (e_m) = M^j_l N^m_k \delta^l_m = M^j_l N^l_k = \delta^j_k\), which means that \(N\) is the inverse of \(M\), \(M^{-1}\).
In fact, as \(M\) is invertible, \(M^{-1}\) exists and is invertible, so, \(B := \{b_j = {M^{-1}}^l_j e_l \vert j \in J\}\) is a basis for \(V\).
\(b^j (b_k) = M^j_l e^l ({M^{-1}}^m_k e_m) = M^j_l {M^{-1}}^m_k e^l (e_m) = M^j_l {M^{-1}}^m_k \delta^l_m = M^j_l {M^{-1}}^l_k = \delta^j_k\), so, \(B\) is indeed a predual of \(B^*\).
\(B\) is unique, because \(N\) needs to be \(M^{-1}\).
For each \(v \in V\), the components of \(v\) with respect to \(B\) are \(b^1 (v), ..., b^d (v)\), because from \(v = v^j b_j\), \(b^l (v) = b^l (v^j b_j) = v^j b^l (b_j) = v^j \delta^l_j = v^l\).
