# sign of the determinant of a transformation in finite dimensional vector space(s)

Why is the definition of the orientation of two bases in a finite dimensional vector space determined by the sign of the determinant of the transformation from one basis to the other?

The definition:

Let V be a finite-dimensional real vector space and let b1 and b2 be two ordered bases for V. It is a standard result in linear algebra that there exists a unique linear transformation A : V → V that takes b1 to b2. The bases b1 and b2 are said to have the same orientation (or be consistently oriented) if A has positive determinant; otherwise they have opposite orientations.

One reason this makes sense is because special orthogonal matrices (with determinant $1$) would only take bases of one orientation to another basis of the same orientation.
So for example in $3$-space , all "right handed" sets of orthonormal vectors can be rotated into all other right handed sets using a (special) orthogonal matrix.