0
$\begingroup$

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.

(copied from http://en.wikipedia.org/wiki/Orientation_(vector_space))

$\endgroup$
0
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.