description/proof of that symmetrization-of-tensor map is linear
Topics
About: vectors space
The table of contents of this article
Starting Context
- The reader knows a definition of symmetrization of tensor with respect to some arguments.
- The reader knows a definition of linear map.
Target Context
- The reader will have a description and a proof of the proposition that any symmetrization-of-tensor map is linear.
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:
\(F\): \(\in \{\text{ the fields }\}\)
\(\{V_1, ..., V_k, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\), where \(V_{j_1} = ... = V_{j_l} := V\) for some \(\{V_{j_1}, ..., V_{j_l}\} \subseteq \{V_1, ..., V_k\}\)
\(L (V_1, ..., V_k: W)\): \(= \text{ the tensors space }\)
\(Sym_{\{j_1, ..., j_l\}}\): \(: L (V_1, ..., V_k: W) \to L (V_1, ..., V_k: W)\), \(= \text{ the symmetrization map }\)
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Statements:
\(Sym_{\{j_1, ..., j_l\}} \in \{\text{ the linear maps }\}\)
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2: Proof
Whole Strategy: Step 1: let \(f_1, f_2 \in L (V_1, ..., V_k: W)\) be any, and see that \(Sym_{\{j_1, ..., j_l\}} (r_1 f_1 + r_2 f_2) = r_1 Sym_{\{j_1, ..., j_l\}} (f_1) + r_2 Sym_{\{j_1, ..., j_l\}} (f_2)\).
Step 1:
Let \(f_1, f_2 \in L (V_1, ..., V_k: W)\) and \(r_1, r_2 \in F\) be any.
\(Sym_{\{j_1, ..., j_l\}} (r_1 f_1 + r_2 f_2) (v_1, ..., v_k) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} (r_1 f_1 + r_2 f_2) (v_{\sigma_1}, ..., v_{\sigma_k})) = 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} (r_1 f_1 (v_{\sigma_1}, ..., v_{\sigma_k}) + r_2 f_2 (v_{\sigma_1}, ..., v_{\sigma_k})) = r_1 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} f_1 (v_{\sigma_1}, ..., v_{\sigma_k}) + r_2 1 / l! \sum_{\sigma \in P_{\{j_1, ..., j_l\}}} f_2 (v_{\sigma_1}, ..., v_{\sigma_k})) = r_1 Sym_{\{j_1, ..., j_l\}} (f_1) (v_1, ..., v_k) + r_2 Sym_{\{j_1, ..., j_l\}} (f_2) (v_1, ..., v_k) = (r_1 Sym_{\{j_1, ..., j_l\}} (f_1) + r_2 Sym_{\{j_1, ..., j_l\}} (f_2)) (v_1, ..., v_k)\).
