Let $T(θ) : \Bbb R^2 → \Bbb R^2$ be the transformation that rotates each vector counterclockwise by angle $θ$.

(a) Write the standard matrix for $T(θ)$.

(b) Explain in words or pictures why $T(θ_1+θ_2) = T(θ_1) ◦ T(θ_2)$.

(c) Derive the angle sum formula for cosine and sine by finding the standard matrix for $T(θ_1+θ_2) = T(θ_1) ◦ T(θ_2)$; that is, prove that $\cos(θ_1+θ_2) = \cos(θ_1)\cos(θ_2)−\sin(θ_1) \sin(θ_2)$ and $\sin(θ_1+θ_2) = \sin(θ_1)\cos(θ_2) + \sin(θ_2)\cos(θ_1)$.

For part a - I'm not sure how to find the standard matrix when I don't know the initial values of the matrix before the vectors rotate.

Part b - I know $T(θ_1) ◦ T(θ_2)$ is the Hadamard product of $T(θ_1)$ and $T(θ_2)$, but I was never taught any properties to prove part b.

Part c - this problem completely confuses me.

Any help would be appreciated.

  • $\begingroup$ As it is implicitly clarified in @Ivo ' s answer, You fix $\theta \in \mathbb{R}$ and take $T(\theta)$ the transform that takes a point $(a,b) \in \mathbb{R}^2$ to $T(\theta)(a,b) \in \mathbb{R}^2$ . And the best way to avoid confusion is to put $\theta$ in the index as Ivo did. For information : the matrix representation will be an element of what we call the special orthogonal group in dimension 2 $\endgroup$ – Aymane Fihadi Nov 11 '15 at 0:27

I'll say what you have to do.

a) Can you compute $T_\theta(1,0)$ and $T_\theta(0,1)$? You'll have to put their coordinates in the columns of the matrix. Make a drawing and see. For example, convince yourself that $T_\theta(1,0) = (\cos \theta, \sin \theta)$.

b) There's nothing to do with Hadamard product, it'll be ordinary matrix multiplication (maps composition). If you rotate by $\theta_1$, and then rotate by $\theta_2$, how much you rotated in the end?

c) Once you have $T_{\theta_1+\theta_2} = T_{\theta_1}\circ T_{\theta_2}$, compute the matrices of both sides and compare each entry. You'll get the formulas simple as that.


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