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Tenshou
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I am curious on how the determinant function determines orientation? I read about in in one of Werner greubs books and I just cannot manage to understand what it is
conquest said:Hi!
The answer before this post seems to be what was asked since it explains exactly how a determinant determines orientation. of course proof is needed here to show that this is well defined, i.e. that the equivalence relation specified by bases are equivalent if the determinant of the change of bases matrix is positive is in fact an equivalence relation.
However the wedge product seems also to be important, maybe to provide some intuition it is a good idea to note here that volume forms are involved. If you try to make a n-linear alternating function on an n-dimensional vectorspace to the base field (which is the natural, if you think about it, definition of a volume form) you will find that up to a constant you will always get the determinant function. So basically every volume form Δ is defined after you know the value of Δ on a basis. Then Δ on any basis is just this constant times the determinant of the matrix with colums the vectors of this basis in terms of the first basis, i.e. the change of basis matrix! So if I calcualte the volume of some n-dimensional 'block' by applying Δ to the vectors that form it the order of the vectors matters! orientation then means that for a matching orientation the volume turns out positive and for the opposite orientation it turns out negative.
In a way then orientation is just a way to deal with things seeming to have negative volume, they only have negative volume if they are oriented against the orientation of the space. Of course this discussion now needs generalization to the case of manifolds. Since then you want to have volume defined in the 'same way' in every tangent space. I am not sure though if anyone wants to discuss that as well.
quasar987 said:An orientation of a vector space V is a choice of a basis B for that space. Once such a choice is made, all the possible basis of V fall into one of two category: either the change of basis matrix from B' to B has positive determinant or negative determinant. In the first case, we then say B' is positively oriented, and in the second case, we say it is negatively oriented.
For instance, on R², the "usual orientation" is to chose B=(e1,e2). Then it is relatively easy to see that according to this definition, the positively oriented basis are the ones (v1,v2) such that the shortest way to rotate v1 onto v2 is anti-clockwise, and the negatively oriented basis are the ones (v1,v2) such that the shortest way to rotate v1 onto v2 is clockwise.
Similarly, in R³, the "usual orientation" is the one specified by (e1,e2,e3). Then you can check that the positively oriented basis are the so called "right handed ones", if you know what that means.
chiro said:Hey Tenshou.
Are you familiar with the cross product?
Once you look at that, you can look at what is called the wedge product and finally its relationship to determinants.
This approach is a lot more intuitive since you can rectify the nature of the cross product geometrically and also see that the cross product is non-commutative since a x b = -(b x a)
Tenshou said:So, when you get the determinant's value it will be positive or negative with respect to the "usual basis/orientation", or zero is they are the same orientation ?
Tenshou said:I am curious on how the determinant function determines orientation? I read about in in one of Werner greubs books and I just cannot manage to understand what it is
A determinant function is a mathematical function that takes in a matrix as its input and outputs a scalar value. It is used to determine whether the matrix is invertible or not, and to calculate the area or volume of a geometric shape.
A determinant function works by using a specific formula to calculate the determinant of a matrix. This formula involves multiplying the elements of the matrix and adding or subtracting them in a specific pattern. The resulting value is the determinant of the matrix.
Understanding orientation is important because it allows us to determine the direction and position of objects in space. This is crucial in many scientific and mathematical applications, such as navigation, mapping, and computer graphics.
A determinant function is a fundamental concept in linear algebra and is used in various ways. It can be used to solve systems of linear equations, find the inverse of a matrix, and determine the rank and eigenvalues of a matrix.
Determinant functions have many real-world applications, such as in physics, engineering, and economics. They are used to calculate the moment of inertia of objects, determine the stability of structures, and analyze the equilibrium of economic systems.