I was watching 3b1b's videos on linear algebra. I cam across following points;
- Negative determinant means orientation of space is reversed. If you assign dimensions to your fingers and if after transformation, if those assignments still hold, then it means orientation of space is not changed and Determinant is positive.
- If after transformation the assignment hold on another hand, then space orientation is reversed and it means Determinant is negative.
My doubts
Q1. Can there be only two assignments, one to left hand and other to right hand?
Q2. If no, then can other assignments also mean reversal of space orientation and negative determinant? That is, after transformation, if assignments don’t hold (neither on left hand nor on right hand), then does it mean space orientation is reversed and Determinant is negative?
Best Answer
As others have noted, you can visualise this in $3$ dimensions (or $2$ for that matter), but let's be more general. A length-preserving linear transformation of vectors is of the form $v\mapsto Rv$ with$$v^Tv=(Rv)^TRv=v^T(R^TR)v$$for all $v$, so $R^TR=I$. Taking determinants, $(\det R)^2=1$, i.e. $\det R=\pm1$. The right-handed choices of axes are related by matrices satisfying $\det R=1$, as are the left-handed ones, but to go from one to the other uses a matrix with $\det R=-1$. In particular, $\det R$ changes sign whenever we invert the direction of one axis, or exchange two axes.