I am trying to prove the following theorem from the book of mathematical physics of Sadri Hassani
Proposition 2.6.9: Let $\Delta$ be a determinant function in the N- Dimensional vector space V. Let $|v\rangle$ and $\{|v_{k}\rangle \}_{k=1}^{N}$ be vectors in V. Then:
$$\sum_{j=1}^{N} (-1)^{(j-1)} \Delta (|v\rangle, |v_{1}\rangle,..., |v_{j}^{*} \rangle,..., |v_{N} \rangle)|v_{j}\rangle= \Delta ( |v_{1}\rangle,..., |v_{N} \rangle) |v\rangle$$
Where the * on a vector mean that it is missing.
I already know how to prove the proposition when i consider that $\{v_{k}\}_{k=1}^{N}$ are linearly independent. but i have not been able to prove that when it is linearly dependent, both the LHS and the RHS are 0. The RHS one is easy to see, as a multilinear functional is 0 when the set is linearly dependent.
However i have been stuck with LHS, any help or advice is well received.