A specific vector transformation processes a vector in adjacent pairs. For each pair `(v_i, v_{i+1})`, it produces a new pair `(-v_{i+1}, v_i)`. If this transformation is applied twice in a row to an input vector `x`, what is the resulting vector in terms of `x`? Justify your answer by showing the steps for a generic 2-dimensional vector `[a, b]`.

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In the element-wise formulation of Rotary Positional Embeddings (RoPE), a permuted version of the input vector $\mathbf{x}$ is required. This vector is constructed by swapping each pair of adjacent elements $(x_{2k-1}, x_{2k})$ and negating the first element of the swapped pair, resulting in $(-x_{2k}, x_{2k-1})$. For a full d-dimensional vector $\mathbf{x}$, the permuted vector is given by: $$ \begin{pmatrix} -x_2 \\ x_1 \\ \vdots \\ -x_d \\ x_{d-1} \end{pmatrix} $$ This vector is a key component in the calculation, where it is multiplied element-wise with the sine vector.

Permuted Input Vector in RoPE

A specific vector transformation operates on a d-dimensional vector by processing it in pairs. For each adjacent pair of elements `(v_{2k-1}, v_{2k})`, the elements are swapped, and then the first element of the new pair is negated. Given an input vector `x = [10, 20, 30, 40]`, what is the resulting vector after this transformation is applied?

A specific vector transformation operates by taking adjacent pairs of elements `(v_{2k-1}, v_{2k})`, swapping them, and negating the first element of the new pair to produce `(-v_{2k}, v_{2k-1})`. If the result of this transformation on an unknown vector `x` is the vector `y = [5, -1, 15, -10]`, what was the original vector `x`?

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