Give the closed-form formula for the wedge product $$ \vec{e}_1\,\wedge\,\vec{e}_2\,\wedge\,\cdots\,\wedge\,\vec{e}_{k} $$as a multilinear form acting on $$ \left(\mathbb{R}^n\right)^k $$ and returning a real number in $$ \mathbb{R}\text{.} $$

The wedge product of the basis vectors $ \vec{e}_1, \vec{e}_2, \dots, \vec{e}_k $ can be represented as a multilinear form acting on $ (\mathbb{R}^n)^k $ and returning a real number. Specifically, the wedge product $ \vec{e}_1 \wedge \vec{e}_2 \wedge \cdots \wedge \vec{e}_k $ is an alternating multilinear form, and when it acts on $ k $ vectors $ \vec{v}_1, \vec{v}_2, \dots, \vec{v}_k \in \mathbb{R}^n $, the result is a determinant.

The closed-form formula is:

\[
\left(\vec{e}_1 \wedge \vec{e}_2 \wedge \cdots \wedge \vec{e}_k\right)(\vec{v}_1, \vec{v}_2, \dots, \vec{v}_k) = \det \left( \begin{array}{ccc}
v_{11} & \dots & v_{1k} \\
v_{21} & \dots & v_{2k} \\
\vdots & & \vdots \\
v_{k1} & \dots & v_{kk} \end{array} \right),
\]
where:

- $ v_{ij} $ is the $ j $-th component of the vector $ \vec{v}_i $.
- The determinant is taken of the $ k \times k $ matrix formed by the components of the vectors $ \vec{v}_1, \vec{v}_2, \dots, \vec{v}_k $.

### Interpretation:
- **Multilinearity:** This form is multilinear, meaning that it is linear in each of the $ \vec{v}_i $.
- **Alternating Property:** The wedge product is alternating, so if any two vectors $ \vec{v}_i $ and $ \vec{v}_j $ are the same, the determinant (and thus the wedge product) will be zero.

### Example:
If $ k = n $ and $ \vec{v}_1, \vec{v}_2, \dots, \vec{v}_n $ are vectors in $ \mathbb{R}^n $, the wedge product $ \vec{e}_1 \wedge \vec{e}_2 \wedge \cdots \wedge \vec{e}_n $ acting on these vectors returns the volume of the parallelepiped spanned by them, given by the determinant of the $ n \times n $ matrix formed by the components of these vectors.

Would you like more information on the properties of this form or examples of its use?

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Here are some related questions you might find interesting:

1. What is the relationship between the wedge product and the determinant?
2. How does the wedge product behave under linear transformations?
3. Can you explain the geometric meaning of the wedge product in $ \mathbb{R}^3 $?
4. What happens when $ k > n $ in the wedge product?
5. How does the wedge product relate to the concept of orientation in geometry?

**Tip:** When working with wedge products, it's helpful to remember that the determinant captures both the volume and the orientation of the parallelepiped formed by the vectors.

Understand the closed-form formula for the wedge product in linear algebra, acting on vectors in $ (\mathbb{R}^n)^k $ and returning a real number. Learn about its multilinear properties and its use in determining determinants of matrices formed by vectors. Suitable for advanced students and professionals in mathematics.

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