The Determinant of a Transition Matrix

Question

Suppose we have a linear map $T$ whose matrix is $A$ such that $$A: V \rightarrow V$$ where $A$ is the transition matrix from the basis $v_1,v_2,...,v_n$ to the basis $w_1,w_2,...,w_n$.

If the determinant of $A$ is positive, then the orientation of the basis $v_1,v_2,...,v_n$ if the same as $w_1,w_2,...,w_n$. If negative, then they have opposite orientations.

Can someone explain why this is? Why are two bases oriented the same if the transition matrix has a positive determinant? What does it mean for two bases to be of the same orientation?

Answer 1

It's basically the definition of ‘same orientation’ and ‘opposite orientation’. To change basis between two bases of same orientation (in $\mathbb{R}^n$), you need to perform a rotation (i.e. orthogonal matrix) and a scaling/shearing (i.e. positive-definite matrix). If, however, you want to change between two bases of different orientations, you have to perform a reflexion as well. Think of it like this: every basis that is oriented differently is just a basis oriented equally with two swapped basis vectors, just as every invertible matrix of negative determinant is just an invertible matrix of positive determinant with two rows swapped

Answer 2

Think you are rotating the coordinate system. $A\in \mathbb{SO}(d)$, so the determinant is 1. If the scaling is also there then the determinant is positive but may not be 1.

On the other hand, you require a reflection, that is why the determinant is negative.