Linear Transformation Question

**r2dee6** · March 3rd 2009, 12:59 PM

Let

is the linear transformation corresponding to a rotation by an angle of

about the x -axis in

and let Let

is the linear transformation corresponding to a rotation by an angle of

about the z -axis

For what values of

and

does

**clic-clac** · March 4th 2009, 01:51 PM

Hi

Rotations are invertible, their inverse being the rotation by the opposite angle about the same axis.
So $T_2.T_1=T_1.T_2\ \Leftrightarrow\ T_1^{-1}.T_2.T_1=T_2$

Observe the action on $x$ of both maps. $\mathbb{R}x$ is left invariant by $T_1$ and $T_1^{-1}$ and is the only subset which has that property.

Thus a necessary condition to the equality is $T_1^{-1}(T_2((x))=T_2(x)\ \text{i.e.}\ T_2(x)\in \mathbb{R}x,$ which means $\vartheta_2=0$ or $\vartheta_2=\pi.$

A similar proof gives: $\vartheta_1=0$ or $\vartheta_1=\pi.$

Therefore a necessary condition is: $((\vartheta_1=0\ \text{or}\ \vartheta_1=\pi)\ \text{and}\ (\vartheta_2=0\ \text{or}\ \vartheta_2=\pi))$

The only thing to show now is that in theese cases, $T_1$ and $T_2$ commute; so the condition will also be sufficient and thus equivalent to the equality.

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