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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?

14tim4
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  • Can you give the definition of orientation you are using? This might help both us and you to see a solution for your problem. – Dirk Sep 13 '17 at 10:06
  • Say the standard basis $(i,j,k)$ is positively oriented because of the right hand rule where k is the thumb, fingers curling from i to j. Sorry, I'm not really too sure, I hope this is what you are looking for to answer my question. – 14tim4 Sep 13 '17 at 10:10

2 Answers2

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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.

Rajat
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  • This might be silly, but why does a transformation that is a reflection give a negative determinant? – 14tim4 Sep 13 '17 at 10:29
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    If you want to learn more about this, then you have to study yourself. For your convenience, check this one:

    https://math.stackexchange.com/questions/68119/why-does-ata-i-det-a-1-mean-a-is-a-rotation-matrix

     – Rajat Sep 13 '17 at 18:47
  • Thanks! I'll have a look – 14tim4 Sep 13 '17 at 21:35
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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

philip98
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