Case of an Euclidean vector plane

[south pilot]

I'm trying to prove the following Proposition and Corollary (Pic), but I don't know how to start.
Can anyone help, please?

[night heart]

Since this is supposed to be the same for any basis, you can take two basis and show that the angles are the same

[south pilot]

the angles?

[night heart]

I meant theta

[lunar frost]

The proposition follows just from taking the four elements of a matrix to be arbitrary and solving the equations resulting from the definition of orthogonality. Then the corollary follows from the fact that the orientation forces determinant to be 1 and now you can solve for the angle since cosine and sine are surjective to [-1, 1] where all the elements of the matrix are too.

[south pilot]

.close