definition of antisymmetric-tensors space w.r.t. field and \(k\) same vectors spaces and vectors space over field
Topics
About: vectors space
The table of contents of this article
Starting Context
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.
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:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V, W\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\( L (V, ..., V: W)\): \(= \text{ the tensors space }\), where \(V\) appears \(k\) times
\(*\Lambda_k (V: W)\): \(= \{t \in L (V, ..., V: W) \vert t \in \{\text{ the antisymmetric-tensors }\}\}\), \(\in \{\text{ the } F \text{ vectors spaces }\}\)
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Conditions:
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\(\Lambda_k (V: W)\) is called "the \(k\)-antisymmetric-tensors space of \(V\) into \(W\)".
\(\Lambda_k (V: F)\) is called "the \(k\)-covectors space of \(V\)".
Any element of \(\Lambda_k (V: F)\) is called "\(k\)-covector".
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 \(\Lambda_k (V: W)\) is indeed an \(F\) vectors space.
1) for any elements, \(t_1, t_2 \in \Lambda_k (V: W)\), \(t_1 + t_2 \in \Lambda_k (V: W)\) (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, \(t_1, t_2 \in \Lambda_k (V: W)\), \(t_1 + t_2 = t_2 + t_1\) (commutativity of addition): it holds in the ambient \(L (V, ..., V; W)\).
3) for any elements, \(t_1, t_2, t_3 \in \Lambda_k (V: W)\), \((t_1 + t_2) + t_3 = t_1 + (t_2 + t_3)\) (associativity of additions): it holds in the ambient \(L (V, ..., V; W)\).
4) there is a 0 element, \(0 \in \Lambda_k (V: W)\), such that for any \(t \in \Lambda_k (V: W)\), \(t + 0 = t\) (existence of 0 vector): the \(0\) map, \(t_0 \in L (V, ..., V: W)\), is in \(\Lambda_k (V: W)\), 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, \(t \in \Lambda_k (V: W)\), there is an inverse element, \(t' \in \Lambda_k (V: W)\), such that \(t' + t = 0\) (existence of inverse vector): \(t' := - t \in L (V, ..., V: W)\); \((- t) (v_{\sigma_1}, ..., v_{\sigma_k}) = - (t (v_{\sigma_1}, ..., v_{\sigma_k})) = - (sgn \sigma t (v_1, ..., v_k)) = sgn \sigma (- t (v_1, ..., v_k)) = sgn \sigma (- t) (v_1, ..., v_k)\), so, \(- t \in \Lambda_k (V: W)\); \(- t + t = 0\) holds because it holds on the ambient \(L (V, ..., V: W)\).
6) for any element, \(t \in \Lambda_k (V: W)\), and any scalar, \(r \in F\), \(r . t \in \Lambda_k (V: W)\) (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 \(r . t \in \Lambda_k (V: W)\).
7) for any element, \(t \in \Lambda_k (V: W)\), 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, \(t_1, t_2 \in \Lambda_k (V: W)\), 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, \(t \in \Lambda_k (V: W)\), 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, \(t \in \Lambda_k (V: W)\), \(1 . t = t\) (identity of 1 multiplication): it holds in the ambient \(L (V, ..., V; W)\).
