# Rewriting determinant in proof of existence of permutation of bases.

I am currently reviewing linear algebra from Steven Roman's Advanced Linear Algebra, and I have gotten stuck on problem 1.23 for quite some time. The question asks given bases $$B = \{b_1, ..., b_n\}, C = \{c_1,...,c_n\}$$ for a vector space $$V$$ and some $$1 \leq m \leq n-1$$, prove there exist some permutation $$\pi$$, such that $$\{b_1, ..., b_m, c_{\pi(m+1)}, ..., c_{\pi(n)}\} \text{ and } \{c_{\pi(1)}, ..., c_{\pi(m)}, b_{m+1}, ..., b_n\}$$ are both bases for $$V$$.

I tried this problem for a few hours, then found a solution here https://staff.math.su.se/mleites/books/prasolov-1994-problems.pdf (Theorem 7.2). However, in this solution, they rewrite the form of the determinant

$$M(b_1, ..., b_n) = \sum_{A \subset V} \pm M(b_1,...,b_m, A)M(V\setminus A, b_{m+1}, ..., b_n)$$

Where $$M(x_1,...,x_n)$$ is the determinant of a matrix with rows $$x_i$$ written in coordinates with respect to the basis $$C$$. I am wondering what identity/theorem is applied here to rewrite the determinant in this way?

• It's the generalized Laplace expansion in the first $k$ rows, using an auxiliary identity for minor determinants. But you don't need to use something so fancy. May 14 at 5:15