1112: Antisymmetric Tensors Space w.r.t. Field and k Same Vectors Spaces and Vectors Space over Field

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definition of antisymmetric tensors space w.r.t. field and $k$ same vectors spaces and vectors space over field

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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: Note

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Starting Context

• The reader knows a definition of tensors space with respect to field and $k$ vectors spaces and vectors space over field.

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Target Context

• The reader will have a definition of antisymmetric tensors space with respect to field and $k$ same vectors spaces and vectors space over field.

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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:
$F$: $\in \{\text{ the fields }\}$
$\{V,W\}$: $\subseteq \{\text{ the }F\text{ vectors spaces }\}$
$L(V,...,V:W)$: $=\text{ the tensors space }$, where $V$ appears $k$ times
$\ast {\mathrm{\Lambda }}_k(V:W)$: $=\{t\in L(V,...,V:W)|t\in \{\text{ the antisymmetric tensors }\}\}$, $\in \{\text{ the }F\text{ vectors spaces }\}$
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Conditions:
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${\mathrm{\Lambda }}_k(V:W)$ is called "the $k$-antisymmetric-tensors space of $V$ into $W$".

${\mathrm{\Lambda }}_k(V):={\mathrm{\Lambda }}_k(V:F)$ is called "the $k$-covectors space of $V$".

Any element of ${\mathrm{\Lambda }}_k(V)$ is called "$k$-covector".

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2: Note

Being antisymmetric means that for any $\sigma \in S^k$ where $S^k$ is the $k$-symmetric group, $t(v_{{\sigma }_1},...,v_{{\sigma }_k})=sgn\sigma t(v_1,...,v_k)$.

All the $\{V_1,...,V_k\}$ are required to be the same $V$, because otherwise, $t(v_{{\sigma }_1},...,v_{{\sigma }_k})$ would not make sense.

Let us see that ${\mathrm{\Lambda }}_k(V:W)$ is indeed an $F$ vectors space.

1) for any elements, $t_1,t_2\in {\mathrm{\Lambda }}_k(V:W)$, $t_1+t_2\in {\mathrm{\Lambda }}_k(V:W)$ (closed-ness under addition): $t_1+t_2\in L(V,...,V:W)$, because $t_j\in L(V,...,V:W)$ and $L(V,...,V:W)$ is an $F$ vectors space; $(t_1+t_2)(v_{{\sigma }_1},...,v_{{\sigma }_k})=t_1(v_{{\sigma }_1},...,v_{{\sigma }_k})+t_2(v_{{\sigma }_1},...,v_{{\sigma }_k})=sgn\sigma t_1(v_1,...,v_k)+sgn\sigma t_2(v_1,...,v_k)=sgn\sigma (t_1(v_1,...,v_k)+t_2(v_1,...,v_k))=sgn\sigma (t_1+t_2)(v_1,...,v_k)$.

2) for any elements, $t_1,t_2\in {\mathrm{\Lambda }}_k(V:W)$, $t_1+t_2=t_2+t_1$ (commutativity of addition): it holds in the ambient $L(V,...,V;W)$.

3) for any elements, $t_1,t_2,t_3\in {\mathrm{\Lambda }}_k(V:W)$, $(t_1+t_2)+t_3=t_1+(t_2+t_3)$ (associativity of additions): it holds in the ambient $L(V,...,V;W)$.

4) there is a 0 element, $0\in {\mathrm{\Lambda }}_k(V:W)$, such that for any $t\in {\mathrm{\Lambda }}_k(V:W)$, $t+0=t$ (existence of 0 vector): the $0$ map, $t_0\in L(V,...,V:W)$, is in ${\mathrm{\Lambda }}_k(V:W)$, because $t_0(v_{{\sigma }_1},...,v_{{\sigma }_k})=0=sgn\sigma t_0(v_1,...,v_k)$, and $t+0=t$ holds because it holds on the ambient $L(V,...,V:W)$.

5) for any element, $t\in {\mathrm{\Lambda }}_k(V:W)$, there is an inverse element, $t^{\prime }\in {\mathrm{\Lambda }}_k(V:W)$, such that $t^{\prime }+t=0$ (existence of inverse vector): $t^{\prime }:=-t\in L(V,...,V:W)$; $(-t)(v_{{\sigma }_1},...,v_{{\sigma }_k})=-(t(v_{{\sigma }_1},...,v_{{\sigma }_k}))=-(sgn\sigma t(v_1,...,v_k))=sgn\sigma (-t(v_1,...,v_k))=sgn\sigma (-t)(v_1,...,v_k)$, so, $-t\in {\mathrm{\Lambda }}_k(V:W)$; $-t+t=0$ holds because it holds on the ambient $L(V,...,V:W)$.

6) for any element, $t\in {\mathrm{\Lambda }}_k(V:W)$, and any scalar, $r\in F$, $r.t\in {\mathrm{\Lambda }}_k(V:W)$ (closed-ness under scalar multiplication): $r.t\in L(V,...,V:W)$; $r.t(v_{{\sigma }_1},...,v_{{\sigma }_k})=r(t(v_{{\sigma }_1},...,v_{{\sigma }_k}))=r(sgn\sigma t(v_1,...,v_k))=sgn\sigma r(t(v_1,...,v_k))=sgn\sigma (r.t)(v_1,...,v_k)$, which means that $r.t\in {\mathrm{\Lambda }}_k(V:W)$.

7) for any element, $t\in {\mathrm{\Lambda }}_k(V:W)$, and any scalars, $r_1,r_2\in F$, $(r_1+r_2).t=r_1.t+r_2.t$ (scalar multiplication distributability for scalars addition): it holds in the ambient $L(V,...,V;W)$.

8) for any elements, $t_1,t_2\in {\mathrm{\Lambda }}_k(V:W)$, and any scalar, $r\in F$, $r.(t_1+t_2)=r.t_1+r.t_2$ (scalar multiplication distributability for vectors addition): it holds in the ambient $L(V,...,V;W)$.

9) for any element, $t\in {\mathrm{\Lambda }}_k(V:W)$, and any scalars, $r_1,r_2\in F$, $(r_1{r}_2).t=r_1.(r_2.t)$ (associativity of scalar multiplications): it holds in the ambient $L(V,...,V;W)$.

10) for any element, $t\in {\mathrm{\Lambda }}_k(V:W)$, $1.t=t$ (identity of 1 multiplication): it holds in the ambient $L(V,...,V;W)$.

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References

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