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Leo-physics
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How to prove any orthogonal transformation can be represented by the product of many mirror transformations, please?What's the intuitive meaning of this proposition?
Thank you.
Thank you.
Def:A ∈L(V,V) V is Euclidean space ,dim V=nSimon Bridge said:You prove it by comparing the definitions, and writing an expression relating the two. You know, how you normally go about proving something in mathematics.
Start by writing out the mathematical expression for a mirror transformation, and also for an orthogonal transformation. What are they, how do they work? How general do you need the proof?
There does not need to be an intuitive idea embodied in a proposition. However, if there is one, it should emerge from the definitions of the terms.
If you have it for ##n=2## have you considered to do it by induction? It would help to see how you've done it.Leo-physics said:When n=1 and n=2 it can be proved, then I am confused about higher dimension
Ah I see, so the mirror transformation s are reflections? I think I remember orthogonal transformation s we're generated by shear maps. Is that what these mirror maps are?Simon Bridge said:As above ... and it can help to pick a specific orthogonal transformation and see what happens, so you get a feel for how the transformations work.
The "intuitive" principle you are exploiting is that the reflection of a reflection is the right way around.
If you prefer, you can put a 1:1 scale image anywhere, in any orientation you like, by use of strategically placed mirrors.
An orthogonal transformation is a type of linear transformation that preserves the length and angle of all vectors in a vector space. It is also known as an isometry because it preserves the shape and size of objects in a space.
An orthogonal transformation is a type of transformation that preserves the length and angle of vectors, while a mirror transformation is a type of transformation that flips an object across a line or plane. While both types of transformations involve changes in orientation, an orthogonal transformation does not change the handedness of an object, while a mirror transformation does.
Orthogonal transformations have many real-life applications, including in computer graphics, engineering, physics, and signal processing. In computer graphics, they are used to rotate, scale, and translate objects. In engineering, they are used to design structures that can withstand forces from different directions. In physics, they are used to describe the motion and orientation of objects in space. In signal processing, they are used to compress and encrypt data.
Yes, an orthogonal transformation can be represented by a matrix. In fact, all orthogonal transformations can be represented by an orthogonal matrix, which is a square matrix with columns and rows that are orthogonal unit vectors. The columns of an orthogonal matrix form an orthonormal basis for the vector space, and the matrix can be used to perform the transformation on vectors and objects in the space.
A mirror transformation affects the determinant of a matrix by changing its sign. If the matrix is reflected across a plane, the determinant will be multiplied by -1. This is because a mirror transformation involves a change in orientation, which can be represented by a negative determinant. In contrast, an orthogonal transformation does not change the determinant of a matrix since it preserves the orientation of the space.