# Math Help - Linear Transformation Question

1. ## Linear Transformation Question

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

2. 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.