how to evaluate $T_2(T_1(u))$.

Question

Let $T_1$ be the linear transformation corresponding to a counterclockwise rotation of $120$ degrees and let $T_2$ be the linear transformation corresponding to a clockwise rotation of $45$ degrees.

Let $u = \begin{bmatrix} 4 \\ 0 \end{bmatrix}$. Then how to evaluate $T_2(T_1(u))$?

My work

I used this formula $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$.

Can you tell me whether this was a correct approach

Answer 1

Guide:

Yes, you can use the formula.

Let the rotation matrix that corresponds to angle $\theta$ counterclockwise to be $A_{\theta}$, you can either compute $A_{\theta_2} (A_{\theta_1}u)$ explicitly.

Or

You can first think of counterclockwise rotation of $120^\circ$ and clockwise rotation of $45^\circ$, compute the net rotation $\theta_3$, and then compute $A_{\theta_3}u=4\begin{bmatrix} \cos \theta_3 \\ \sin \theta_3\end{bmatrix}.$