Permuted Input Vector in RoPE

A transformation is applied to a d-dimensional vector $\mathbf{x}$ at position $t$ using a set of frequencies $\theta$. The transformation is defined by the following element-wise formula: $\mathrm{Trans}(\mathbf{x}, t\theta) = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_{d-1} \\ x_d \end{pmatrix} \odot \begin{pmatrix} \cos t\theta_1 \\ \cos t\theta_1 \\ \vdots \\ \cos t\theta_{d/2} \\ \cos t\theta_{d/2} \end{pmatrix} + \begin{pmatrix} -x_2 \\ x_1 \\ \vdots \\ -x_d \\ x_{d-1} \end{pmatrix} \odot \begin{pmatrix} \sin t\theta_1 \\ \sin t\theta_1 \\ \vdots \\ \sin t\theta_{d/2} \\ \sin t\theta_{d/2} \end{pmatrix}$ where $\odot$ denotes element-wise multiplication.

Given the input vector $\mathbf{x} = [1, 2, 3, 4]$, position $t=1$, and frequencies $\theta_1 = \pi/2$, $\theta_2 = \pi$, what is the resulting vector $\mathrm{Trans}(\mathbf{x}, t\theta)$?

Analyzing Components of a Vector Transformation

Consider the element-wise formula for applying a rotational transformation to a d-dimensional vector $\mathbf{x}$. The formula involves summing two vectors derived from element-wise products.

True or False: The first vector in the sum is the product of the original input vector $\mathbf{x}$ and a sine vector, and the second vector is the product of a permuted input vector (e.g., $[-x_2, x_1, \dots]$) and a cosine vector.