I am having troubles understanding this proof. I don't understand how, if $\eta \in \Lambda^n(\mathbb{R^n})$ implies that $\eta = \lambda \cdot \det$ - really I don't see how determinants come from this.
This is theorem 4-6 from spivak.
2026-03-28 20:10:25.1774728625
Alternating k-tensors and determinant (theorem 4-6, Spivak)
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This follows from the fact that $\bigwedge^n (T^*M)$ (in other words, the space of $n$-forms on an $n$-dimensional manifold) is 1-dimensional. Since the determinant is one such form, all others are scalar multiples of it.