# Unique volume element in a vector space

by yifli
Tags: element, space, unique, vector, volume
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 P: 70 Given two orthonormal bases $v_1,v_2,\cdots,v_n$ and $u_1,u_2,\cdots,u_n$ for a vector space $V$, we know the following formula holds for an alternating tensor $f$: $$f(u_1,u_2,\cdots,u_n)=\det(A)f(v_1,v_2,\cdots,v_n)$$ where A is the orthogonal matrix that changes one orthonormal basis to another orthonormal basis. The volume element is the unique $f$ such that $f(v_1,v_2,\cdots,v_n)=1$ for any orthonormal basis $v_1,v_2,\cdots,v_n$ given an orientation for $V$. Here is my question: If $\det(A)=-1$, then $f(u_1,u_2,\cdots,u_n)=-1$, which makes $f$ not unique
 Mentor P: 18,019 Hi yifli! I don't quite see why your reasoning implies that f is not unique. What's the other tensor that makes $f(v_1,...,v_n)=1$?
P: 70
 Quote by yifli Here is my question: If $\det(A)=-1$, then $f(u_1,u_2,\cdots,u_n)=-1$, which makes $f$ not unique
 Quote by micromass Hi yifli! I don't quite see why your reasoning implies that f is not unique. What's the other tensor that makes $f(v_1,...,v_n)=1$?
Sorry for the confusion. Actually I mean $f(u_1,u_2,\cdots,u_n)$ is not necessarily equal to 1, it may be equal to -1 also, even though $f(v_1,v_2,\cdots,v_n)$ is made to be 1

 Mentor P: 18,019 Unique volume element in a vector space Yes, but the point is that f of every basis with the same orientation will give you 1. If you take the opposite orientation, then you will get -1 of course.
P: 70
 Quote by micromass Yes, but the point is that f of every basis with the same orientation will give you 1. If you take the opposite orientation, then you will get -1 of course.
So given a fixed orientation, it's impossible for $\det(A)$ to be -1?

What does it mean for $\det(A)$ to be -1?
Mentor
P: 18,019
 Quote by yifli So given a fixed orientation, it's impossible for $\det(A)$ to be -1? What does it mean for $\det(A)$ to be -1?
The transition matrix between two orthonormal bases has det(A)=1 if and only if the bases have the same orientation and has det(A)=-1 if they have opposite orientation.
 HW Helper P: 6,187 A orthonormal transformation matrix with det(A)=1 is a rotation. A orthonormal transformation matrix with det(A)=-1 is a rotation combined with a reflection. So the volume sign would be invariant.

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