- #1

- 2

- 0

Thank you.

- I
- Thread starter Leo-physics
- Start date

- #1

- 2

- 0

Thank you.

- #2

Simon Bridge

Science Advisor

Homework Helper

- 17,857

- 1,654

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.

- #3

- 2

- 0

Def:

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.

Def:

Question:

Prove: If A is an orthogonal transformation over V (Euclidean space)

⇒∀α∈V

(Note : When n=1 and n=2 it can be proved, then I am confused about higher dimension )

- #4

fresh_42

Mentor

- 13,817

- 10,986

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.When n=1 and n=2 it can be proved, then I am confused about higher dimension

- #5

WWGD

Science Advisor

Gold Member

2019 Award

- 5,410

- 3,487

- #6

Simon Bridge

Science Advisor

Homework Helper

- 17,857

- 1,654

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.

- #7

WWGD

Science Advisor

Gold Member

2019 Award

- 5,410

- 3,487

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?

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.

Last edited:

- #8

Simon Bridge

Science Advisor

Homework Helper

- 17,857

- 1,654

- #9

- 177

- 61

if you know how to prove this for a rotation in ##\mathbb R^2##, then you can get the desired statement from results in s.5 of Chapter 6 of "Linear Algebra Done Wrong", see Theorems 5.1, 5.2 there.

- Replies
- 9

- Views
- 5K

- Last Post

- Replies
- 11

- Views
- 7K

- Last Post

- Replies
- 1

- Views
- 7K

- Last Post

- Replies
- 4

- Views
- 2K

- Replies
- 3

- Views
- 5K

- Replies
- 4

- Views
- 2K

- Replies
- 2

- Views
- 1K

- Replies
- 2

- Views
- 7K

- Last Post

- Replies
- 3

- Views
- 2K

- Replies
- 3

- Views
- 1K