In the Euclidean space $\mathbb{R}^n$ consider two (ordered) sets of vectors $a_1 \ldots a_k$ and $b_1 \ldots b_k$ with $k \le n$.
Question
- What is the geometrical interpretation of $\det(a_i \cdot b_j)?$
- Is it true that $\det(a_i\cdot b_j)=\det(a'_p\cdot b'_q)$ if $a_1\wedge \ldots \wedge a_k=a'_1\wedge \ldots \wedge a'_k$ and $b_1\wedge \ldots \wedge b_k=b'_1\wedge \ldots \wedge b'_k?$
Since $\det(a_i\cdot a_j)$ equals the squared $k$-volume spanned by $a_1\ldots a_k$, I guess that $\det(a_i\cdot b_j)$ may be interpreted as the $k$-volume spanned by some kind of projection of $b_1\wedge \ldots \wedge b_k$ (thought of as an oriented $k$-parallelogram) onto $a_1\wedge \ldots \wedge a_k$. For the same reason I would answer affirmatively to the second question.