Verifying Equivalence of 2D Rotational Methods
Your task is to algebraically demonstrate that the two methods described below for rotating a 2D vector are equivalent. Show your work by expanding the complex multiplication in Method A and proving that its resulting real and imaginary components are identical to the formulas for the new vector components in Method B.
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Ch.3 Prompting - Foundations of Large Language Models
Foundations of Large Language Models
Foundations of Large Language Models Course
Computing Sciences
Application in Bloom's Taxonomy
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Euclidean Representation of RoPE Rotation
Element-wise Formula for RoPE Rotation
A positional encoding method can be described in two ways. In one view, a d-dimensional vector is treated as d/2 complex numbers, and each complex number is rotated by an angle. In another view, the same d-dimensional vector is treated as d/2 pairs of real numbers, and each pair is rotated as a 2D vector by the same corresponding angle. What is the core principle that makes these two views equivalent?
Equivalence of 2D Vector Rotation Representations
Verifying Equivalence of 2D Rotational Methods