Let $$ \mathbb{F} \in \left\{\mathbb{R}, \mathbb{C}\right\} $$, let $$ \left(n,k\right) $$ be an ordered pair in $$ \mathbb{N}^2 $$ such that $$ 1\le k\le n $$, let $$ \left(V,\text{ }+,\text{ }\cdot,\text{ }\langle\cdot,\cdot\rangle\right) $$ be a $$ n $$-dimensional $$ \mathbb{F} $$-inner product space and let $$ \left(\mathbf{\vec{v}}_{i}\right)_{i=1}^{k} $$ be a $$ k $$-tuple of vectors in $$ V^{k} $$. Give me the rigorous and precise definition of the $$ k $$-vector $$ \bigwedge_{i = 1}^{k}\mathbf{\vec{v}}_{i} $$ as an equivalence class of oriented $$ k $$-dimensional parallelepipeds (using the notion of orientation).

### Definition of the $ k $-Vector $ \bigwedge_{i = 1}^{k} \mathbf{\vec{v}}_i $ as an Equivalence Class of Oriented $ k $-Dimensional Parallelepipeds

Let $ \mathbb{F} \in \{\mathbb{R}, \mathbb{C}\} $, and let $ V $ be an $ n $-dimensional $ \mathbb{F} $-inner product space. Consider the ordered $ k $-tuple $ (\mathbf{\vec{v}}_i)_{i=1}^{k} $ of vectors in $ V $, where $ 1 \leq k \leq n $.

#### Oriented $ k $-Dimensional Parallelepipeds
A $ k $-dimensional parallelepiped in $ V $ is defined as the set of points
\[
P(\mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k) = \left\{ \sum_{i=1}^{k} \lambda_i \mathbf{\vec{v}}_i \mid \lambda_i \in [0, 1] \text{ for all } i \right\},
\]
where the vectors $ \mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k $ are linearly independent. This set defines a $ k $-dimensional parallelepiped spanned by the vectors $ \mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k $.

An *orientation* on this parallelepiped is a choice of an equivalence class of ordered $ k $-tuples $ (\mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k) $ under the relation of permutation with sign:
\[
(\mathbf{\vec{v}}_{\sigma(1)}, \mathbf{\vec{v}}_{\sigma(2)}, \dots, \mathbf{\vec{v}}_{\sigma(k)}) \sim \text{sign}(\sigma) (\mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k),
\]
where $ \sigma $ is a permutation of the indices $ \{1, 2, \dots, k\} $, and $\text{sign}(\sigma)$ denotes the sign of the permutation.

#### The $ k $-Vector $ \bigwedge_{i=1}^{k} \mathbf{\vec{v}}_i $
The $ k $-vector $ \bigwedge_{i=1}^{k} \mathbf{\vec{v}}_i $ is defined as the equivalence class of the oriented $ k $-dimensional parallelepiped spanned by the ordered $ k $-tuple $ (\mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k) $.

Formally, let $ P = P(\mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k) $ be a $ k $-dimensional parallelepiped. The $ k $-vector $ \bigwedge_{i=1}^{k} \mathbf{\vec{v}}_i $ corresponds to the equivalence class $ [P, o] $ of this oriented parallelepiped, where $ o $ denotes the orientation induced by the ordered tuple $ (\mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k) $.

This equivalence class is defined as follows:
- **Equivalence:** Two oriented $ k $-dimensional parallelepipeds $ (P, o) $ and $ (P', o') $ are considered equivalent if there exists a non-zero scalar $ \lambda \in \mathbb{F} $ such that $ P' = \lambda P $ and $ o' $ is the orientation corresponding to the ordered tuple $ (\lambda \mathbf{\vec{v}}_1, \lambda \mathbf{\vec{v}}_2, \dots, \lambda \mathbf{\vec{v}}_k) $.

Thus, the $ k $-vector $ \bigwedge_{i=1}^{k} \mathbf{\vec{v}}_i $ represents the oriented $ k $-dimensional parallelepiped defined by the vectors $ \mathbf{\vec{v}}_1, \mathbf{\vec{v}}_2, \dots, \mathbf{\vec{v}}_k $, up to this equivalence relation.

Would you like more details or have any questions? Here are some related questions:

1. How does the exterior product generalize to non-commutative algebras?
2. What is the geometric significance of the orientation of a parallelepiped?
3. How does the concept of orientation relate to determinants in linear algebra?
4. How are $ k $-vectors used in differential geometry?
5. Can you explain the role of $ k $-vectors in defining differential forms?

**Tip:** Understanding the concept of orientation is crucial when working with exterior products, as it directly influences the sign and properties of the resulting vectors.

Explore the rigorous definition of the k-vector $\bigwedge_{i = 1}^{k}\mathbf{\vec{v}}_{i}$ as an equivalence class of oriented k-dimensional parallelepipeds in inner product spaces. This detailed explanation covers the orientation of parallelepipeds and their significance in vector space theory.

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