I am trying to define the wedge product $$ \bm{\vec{u}_1\wedge\vec{u}_2\wedge\cdots\wedge\vec{u}_{k}} $$of $$ k $$ $$ n $$-dimensional vectors as an alternating multilinear form acting on $$ \left(\mathbb{R}^n\right)^k $$, returning a real number in $$ \mathbb{R} $$. Can you give me the closed-form formula for this wedge product?

The wedge product (also known as the exterior product) of $ k $ vectors in $ \mathbb{R}^n $ is defined as an alternating multilinear form. The wedge product of $ k $ vectors $\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_k $ can be expressed as a determinant in the following closed-form formula:

\[
\vec{u}_1 \wedge \vec{u}_2 \wedge \cdots \wedge \vec{u}_k = \sum_{\sigma \in S_k} \text{sgn}(\sigma) \, u_{1\sigma(1)} u_{2\sigma(2)} \cdots u_{k\sigma(k)} \, \bm{e}_{\sigma(1)} \wedge \bm{e}_{\sigma(2)} \wedge \cdots \wedge \bm{e}_{\sigma(k)},
\]
where:

- $ S_k $ is the symmetric group of all permutations of $ k $ elements.
- $ \text{sgn}(\sigma) $ is the sign (parity) of the permutation $ \sigma $.
- $ u_{i\sigma(i)} $ represents the $ \sigma(i) $-th component of the vector $ \vec{u}_i $.
- $ \bm{e}_1, \bm{e}_2, \ldots, \bm{e}_n $ are the standard basis vectors in $ \mathbb{R}^n $.
  
In simpler terms, if you consider the wedge product of the vectors $\vec{u}_1, \vec{u}_2, \ldots, \vec{u}_k$, it is given by:

\[
\vec{u}_1 \wedge \vec{u}_2 \wedge \cdots \wedge \vec{u}_k = \sum_{\sigma \in S_k} \text{sgn}(\sigma) \det \left( \begin{array}{ccc} u_{1\sigma(1)} & \dots & u_{1\sigma(k)} \\ u_{2\sigma(1)} & \dots & u_{2\sigma(k)} \\ \vdots & & \vdots \\ u_{k\sigma(1)} & \dots & u_{k\sigma(k)} \end{array} \right) \bm{e}_{\sigma(1)} \wedge \bm{e}_{\sigma(2)} \wedge \cdots \wedge \bm{e}_{\sigma(k)},
\]
where the determinant is taken of a $ k \times k $ matrix formed by the components of the vectors.

This expression returns a real number (a scalar) when $ k = n $. The wedge product is a key concept in differential forms and exterior algebra, with important applications in geometry and physics.

Would you like further details on how this formula is derived or its applications?

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Here are some related questions you might be interested in:

1. How does the wedge product differ from the dot product and cross product?
2. What are some applications of the wedge product in differential geometry?
3. Can the wedge product be extended to more abstract vector spaces?
4. How does the wedge product relate to determinants?
5. What is the geometric interpretation of the wedge product in $ \mathbb{R}^3 $ or $ \mathbb{R}^4 $?

**Tip:** Understanding the geometric interpretation of the wedge product can significantly enhance your intuition, especially in higher dimensions.

Understand the wedge product of k n-dimensional vectors in $ \mathbb{R}^n $ with a detailed formula and explanation. This concept is crucial in differential forms and exterior algebra, returning a real number in $ \mathbb{R} $.

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