definition of interior multiplication of tensor by vector
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of interior multiplication of tensor by vector.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_1, ..., V_k, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\( L (V_1, ..., V_k: W)\): \(= \text{ the tensors space }\)
\( L (V_2, ..., V_k: W)\): \(= \text{ the tensors space }\)
\( v\): \(\in V_1\)
\(*i_v\): \(: L (V_1, ..., V_k: W) \to L (V_2, ..., V_k: W), t (\bullet) \mapsto t (v, \bullet)\), \(\in \{\text{ the } F \text{ linear maps }\}\)
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Conditions:
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2: Note
Let us see that \(i_v\) is indeed into \(L (V_2, ..., V_k: W)\).
\(i_v (t): V_2 \times ... \times V_k \to W\).
\(i_v (t) (v_2, ..., r v_j + r' v'_j, ..., v_k) = t (v, v_2, ..., r v_j + r' v'_j, ..., v_k) = r t (v, v_2, ..., v_j, ..., v_k) + r' t (v, v_2, ..., v'_j, ..., v_k) = r i_v (t) (v_2, ..., v_j, ..., v_k) + r' i_v (t) (v_2, ..., v'_j, ..., v_k)\).
So, \(i_v (t) \in L (V_2, ..., V_k: W)\).
Let us see that \(i_v\) is indeed an \(F\) linear map.
\((i_v (r t + r' t')) (v_2, ..., v_k) = (r t + r' t') (v, v_2, ..., v_k) = r t (v, v_2, ..., v_k) + r' t' (v, v_2, ..., v_k)\).
\((r i_v (t) + r' i_v (t')) (v_2, ..., v_k) = (r i_v (t)) (v_2, ..., v_k) + (r' i_v (t')) (v_2, ..., v_k) = r (i_v (t)) (v_2, ..., v_k) + r' (i_v (t')) (v_2, ..., v_k) = r t (v, v_2, ..., v_k) + r' t' (v, v_2, ..., v_k) \).
That means that \(i_v (r t + r' t') = r i_v (t) + r' i_v (t')\).
Usually, interior multiplication of antisymmetric tensor by vector is used, but this definition is not invalid in any way.
