I am supposed to show
$$\beta^1 ∧ ··· ∧ \beta^k = (\det A) \gamma^1 ∧ ··· ∧ \gamma^k$$
for covectors $\beta^1, ···, \beta^k, \gamma^1, ···, \gamma^k$ of a vector space finite dimensional real vector space $V$ of dimension $n$ such that $\vec{\beta}=A\vec{\gamma}$, where
$$\vec{\beta} = [\beta^1 ... \beta^k]^T, \vec{\gamma} = [\gamma^1 ... > \gamma^k]^T, A = [a_1^T ... a_k^T]^T, a_i^T = [a^i_1 ... a^i_k]$$
The solutions I have found (like this one on stackexchange and two more that are not on stackexchange) are not matrix-based and do not seem to utilize the following fact about the wedge product of covectors
$$(\beta^1 ∧ ··· ∧ \beta^k) (v_1, ..., v_k) = \det[\beta^{i}(v_j)]$$
- So I'm composing a matrix-based proof now. Please verify.
Let $v_1, ..., v_k \in V$. Then $$LHS = (\beta^1 ∧ ··· ∧ \beta^k) (v_1, ..., v_k) = \det[\beta^{i}(v_j)] = \det[\vec{\beta}(v_1) ... \vec{\beta}(v_k)],$$ where $\vec{\beta}(v_j) = [\beta^1(v_j) ... \beta^k(v_j)]^T$.
Now,
$$[\vec{\beta}(v_1) ... \vec{\beta}(v_k)] = [A\vec{\gamma}(v_1) ... A\vec{\gamma}(v_k)] = A [\vec{\gamma}(v_1) ... \vec{\gamma}(v_k)] = A [\gamma^i(v_j)]$$
Therefore,
$$LHS = \det[\vec{\beta}(v_1) ... \vec{\beta}(v_k)] = \det (A [\gamma^i(v_j)])= \det (A) \det([\gamma^i(v_j)])$$
$$ = \det(A) (\gamma^1 ∧ ··· ∧ \gamma^k)(v_1, ..., v_k) = RHS$$
My book is An Introduction to Manifolds by Loring W. Tu. This is Exercise 3.7 and is called "Transformation rule for a wedge product of covectors". The above fact is Proposition 3.27 called "wedge product of 1-covectors" (covector is defined as 1-covector).
Motivation for a matrix-based proof: I believe there's a way to do a matrix-based proof for another exercise, Exercise 3.8, based on an analogous result for Proposition 3.27 and Exercise 3.1 on inner product (specifically the generalization of Exercise 3.1 given in Exercise 3.3). If the above proof is unsuccessful, then I might not continue to attempt a matrix-based proof for Exercise 3.8. If the above proof is successful, then I might consider posting my attempt of a matrix-based proof for Exercise 3.8 in another question.
