Can you please help prove or disprove my conjecture below?
Definition: Given vector space $V$ of $n$ dimensions, and ordered lists $x, y$ such that $x,y \subset V$; $x,y$ are each linearly independent; and $x,y$ each span $V$, then $x$ and $y$ are said to be of the same orientation if there exists a linear transformation $R$ such that:
- $R(V) = V$
- $\det R = 1$
- For all $i \leq n$, $y_i$ is in the span of $\{x_1 ... x_i\}$ with the coefficient of $x_i$ nonnegative. Formally, this is: For each $i \leq n$ there exists an ordered list $S_i$ of $i$ scalars $\{S_{i_1}, S_{i_2}, ..., S_{i_i}\}$ with $S_{i_i} \geq 0$ such that $R(x_i) = \sum_{j \leq i} S_{i_j} y_j$.
Conjecture: Let $X$ be a matrix in any basis where column $i$ of $X$ is $x_n$, and let $Y$ be similarly constructed from $y$ in the same basis. Then $x$ and $y$ are in the same orientation if and only if $\det XY > 0$.
Corollaries:
- Same orientation is an equivalence relation
- An even permutation of list $x$ or $y$ preserves sameness of orientation; an odd permutation alters it
- If $a,b$ are linearly independent vectors in $\mathbb R^3$, then $a, b, a \times b$ are in the same orientation as $e_1, e_2, e_3$.
Motivation: The Right Hand Rule is used throughout physics as well as in well-respected vector calculus books, such as Hubbard or Colley. Yet, it's hard if not impossible to formulate the RHR without reference to anatomy. The above (if proven true!) is an attempt to do so in a mathematically rigorous way.
I'd add a fourth "corollary": If a person views themselves as living inside $\mathbb R^3$, with $e_1, e_2, e_3$ drawn as conventional (to the right, forward, and up, respectively), then their right hand's thumb, pointer, and middle finger are in the same orientation as $e_1, e_2, e_3$.
Is that conjecture true? Can you help me prove it? What would $R$ look like? (Point #3 reminds me of a triangular matrix. Perhaps $R$ could be specified as $PTP^{-1}$, where $\det P = 1$ and $T$ is triangular?). Can we find $R$?