Definition of the 2D Rotation Matrix
The 2D rotation matrix, denoted as , is used to perform a counter-clockwise rotation of a vector by an angle in a two-dimensional plane. For post-multiplication with a row vector, the matrix is defined as:
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Ch.2 Generative Models - Foundations of Large Language Models
Foundations of Large Language Models
Foundations of Large Language Models Course
Computing Sciences
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Composition Property of Rotations
Representing 2D Vector Rotation in Complex Space
Definition of the 2D Rotation Matrix
Consider a system where the position of a token in a sequence is encoded by rotating its initial vector embedding,
x. The total angle of rotation is directly proportional to the token's position,m. If the vector for a token at position 3 is obtained by rotatingxby a total angle of3θ, what is the correct transformation to find the vector for the same token at position 9?A positional encoding system represents a token's position by sequentially rotating its initial vector embedding, denoted as
x, by a fixed angleθfor each step forward in a sequence. Arrange the following vector states to show the correct order of transformations for a token as its position advances from 1 to 3.In a system that encodes sequential position, an initial vector
xis transformed to represent positionmby applying a cumulative rotation, resulting in vectorv_m. Similarly, the vector for positionm+1isv_{m+1}. Based on this mechanism, what is the direct geometric transformation that relatesv_mtov_{m+1}?
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Application of RoPE Rotation to a 2D Vector
Definition of the 2x2 RoPE Rotation Matrix Block
A developer is implementing a function to rotate a 2D row vector
vcounter-clockwise by an angleθ. The operation is performed by post-multiplying the vector by a 2x2 matrixR(i.e.,v_rotated = vR). Which of the following matrices correctly represents this transformation?Constructing a 90-Degree Rotation Matrix
Consider a transformation applied to a 2D row vector
vby post-multiplying it with a matrixR(i.e.,v_rotated = vR). The matrix correctly performs a counter-clockwise rotation of the vector by an angle θ.