1017: Antisymmetrization-of-Tensor Map Is Linear

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description/proof of that antisymmetrization-of-tensor map is linear

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Proof

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Starting Context

• The reader knows a definition of antisymmetrization of tensor with respect to some arguments.
• The reader knows a definition of linear map.

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Target Context

• The reader will have a description and a proof of the proposition that any antisymmetrization-of-tensor map is linear.

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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.

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Main Body

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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 $\{V_{j_1},...,V_{j_l}\}\subseteq \{V_1,...,V_k\}$
$L(V_1,...,V_k:W)$: $=\text{ the tensors space }$
$Asy{m}_{\{j_1,...,j_l\}}$: $:L(V_1,...,V_k:W)\to L(V_1,...,V_k:W)$, $=\text{ the antisymmetrization map }$
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Statements:
$Asy{m}_{\{j_1,...,j_l\}}\in \{\text{ the linear maps }\}$
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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 $Asy{m}_{\{j_1,...,j_l\}}(r_1{f}_1+r_2{f}_2)=r_{1}Asy{m}_{\{j_1,...,j_l\}}(f_1)+r_{2}Asy{m}_{\{j_1,...,j_l\}}(f_2)$.

Step 1:

Let $f_1,f_2\in L(V_1,...,V_k:W)$ and $r_1,r_2\in F$ be any.

$Asy{m}_{\{j_1,...,j_l\}}(r_1{f}_1+r_2{f}_2)(v_1,...,v_k)=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}sgn\sigma (r_1{f}_1+r_2{f}_2)(v_{{\sigma }_1},...,v_{{\sigma }_k}))=1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}sgn\sigma (r_1{f}_1(v_{{\sigma }_1},...,v_{{\sigma }_k})+r_2{f}_2(v_{{\sigma }_1},...,v_{{\sigma }_k}))=r_{1}1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}sgn\sigma f_1(v_{{\sigma }_1},...,v_{{\sigma }_k})+r_{2}1/l!\sum _{\sigma \in P_{\{j_1,...,j_l\}}}sgn\sigma f_2(v_{{\sigma }_1},...,v_{{\sigma }_k}))=r_{1}Asy{m}_{\{j_1,...,j_l\}}(f_1)(v_1,...,v_k)+r_{2}Asy{m}_{\{j_1,...,j_l\}}(f_2)(v_1,...,v_k)=(r_{1}Asy{m}_{\{j_1,...,j_l\}}(f_1)+r_{2}Asy{m}_{\{j_1,...,j_l\}}(f_2))(v_1,...,v_k)$.

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References

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