If $A$ is a rotation matrix by $\theta$, then what does $A^T$ do?

Question

Little help here?

If multiplication by $A$ rotates a vector $x$ in the $x$-$y$ plane through an angle $\theta$, then what is the effect of multiplying $x$ by $A^T$. Explain your reasoning.

Any help would be greatly appreciated. Thanks

Answer 1

Hints:

What is the matrix that represents rotation in the xy-plane by angle $\theta$ counterclockwise?

(Googling it or searching MSE will surely get you the matrix, but even better would be for you to try to compute it from scratch, which requires just a bit more thought...)

Now, what is the transpose of this matrix?

Finally, take simple basis vectors of $\mathbb{R}^2$; what does the transpose of the original rotation matrix do to these vectors?

Key point is this: knowing the action of your linear operator (or matrix, with respect to some chosen basis) on basis vectors completely determines the action of the linear operator on all vectors in the vector space, i.e., in the two-dimensional $x$-$y$ plane.

Answer 2

Hints:

• A rotation is an orthogonal transformation.
• An orthogonal matrix $M$ has $M^t=M^{-1}$.