Is your equivalence relation equal to the equivalence relation $$ \simeq $$ defined by:$$  $$
$$ \left(\bm{\vec{u}}_{i}\right)_{i=1}^{k}\simeq\left(\bm{\vec{v}}_{i}\right)_{i=1}^{k}\iff\det\left([\langle\bm{\vec{u}}_{i},\bm{\vec{v}}_{j}\rangle]_{1\leqslanti,j\leqslantn}\right)>0 $$?

No, the equivalence relation I described in my initial response is **not** the same as the equivalence relation $\simeq$ defined by the condition
\[
\left(\bm{\vec{u}}_{i}\right)_{i=1}^{k}\simeq\left(\bm{\vec{v}}_{i}\right)_{i=1}^{k} \iff \det\left([\langle\bm{\vec{u}}_{i},\bm{\vec{v}}_{j}\rangle]_{1\leqslant i,j \leqslant k}\right) > 0.
\]

### Explanation of the Equivalence Relation $\simeq$
The equivalence relation $\simeq$ you described is based on the *positive definiteness* of the determinant of the Gram matrix. Specifically:
- Given two $ k $-tuples of vectors $\left(\bm{\vec{u}}_i\right)_{i=1}^{k}$ and $\left(\bm{\vec{v}}_i\right)_{i=1}^{k}$, the relation $\simeq$ holds if the determinant of the Gram matrix $ [\langle \bm{\vec{u}}_i, \bm{\vec{v}}_j \rangle] $ is positive.
- The Gram matrix $ G = [\langle \bm{\vec{u}}_i, \bm{\vec{v}}_j \rangle] $ is a $ k \times k $ matrix, where each entry is the inner product of corresponding vectors from the two $ k $-tuples.
- The determinant $\det(G)$ measures the "oriented volume" of the parallelepiped spanned by the vectors $ \bm{\vec{u}}_1, \bm{\vec{u}}_2, \dots, \bm{\vec{u}}_k $ relative to the vectors $ \bm{\vec{v}}_1, \bm{\vec{v}}_2, \dots, \bm{\vec{v}}_k $. A positive determinant indicates that the two $ k $-tuples define parallelepipeds with the same orientation.

### Equivalence Relation from the Initial Definition
The equivalence relation I initially described is based on scalar multiplication and orientation, where two $ k $-tuples $ (\bm{\vec{u}}_i)_{i=1}^{k} $ and $ (\bm{\vec{v}}_i)_{i=1}^{k} $ are equivalent if:
\[
\bm{\vec{v}}_i = \lambda \bm{\vec{u}}_i \quad \text{for some non-zero scalar } \lambda \in \mathbb{F}, \text{ and for all } i.
\]
This is different from the equivalence relation $\simeq$, which is more geometric and depends on the positive definiteness of the Gram determinant.

### Summary of Differences
- **Initial Equivalence Relation:** Based on scalar multiplication and orientation (sign of permutation).
- **Equivalence Relation $\simeq$:** Based on the positive definiteness of the determinant of the Gram matrix formed by the inner products of vectors in the two tuples.

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

1. How does the Gram matrix relate to the geometry of vectors?
2. Can you describe the geometric interpretation of the determinant of a Gram matrix?
3. What are the conditions under which two parallelepipeds are considered equivalent in orientation?
4. How does the orientation influence the sign of the determinant in linear algebra?
5. Can you explore the role of inner product spaces in defining equivalence relations like $\simeq$?

**Tip:** When working with vector spaces, the Gram matrix and its determinant are key tools for understanding geometric properties like volume and orientation.

Explore the difference between two equivalence relations in linear algebra: one based on scalar multiplication and orientation, and the other based on the positive definiteness of the determinant of the Gram matrix. Gain insights into the geometric interpretation and mathematical foundations. Ideal for advanced students and professionals in mathematics and linear algebra.

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