Component-wise Vector Rotation in Complex Space
The transformation performs component-wise rotations on a vector in complex space. It is defined by the formula: This equation represents the final rotated vector as a linear combination. Each term in the sum consists of a complex component (derived from the original real vector ) rotated by an angle , and then scaled by its corresponding standard basis vector . The term represents a vector with a value of 1 in the k-th position and zeros elsewhere.
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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
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Formula for Applying RoPE Rotation 't' Times
Definition of C(x, tθ) Notation for Rotated Tokens
Component-wise Vector Rotation in Complex Space
A 2D vector (x, y) can be represented by the complex number z = x + iy. To rotate this vector by an angle θ, it is multiplied by the complex number e^(iθ) = cos(θ) + i sin(θ). If the vector (3, 4) is rotated by 90 degrees (π/2 radians) using this method, what are the coordinates of the new, rotated vector?
A 2D vector can be rotated by representing it as a complex number and multiplying it by another complex number that represents the rotation. Arrange the following steps in the correct logical order to derive the coordinates of a vector
(x₁, x₂)after it has been rotated by an angleθ.Mapping Complex Multiplication to Rotated Vector Components
Component-wise Vector Rotation in Complex Space
Consider a vector
vin a 3-dimensional space, defined asv = [4, -7, 2]. Which of the following options correctly expressesvas a weighted sum of vectors, where each of these vectors has a value of 1 in a single coordinate and 0s in all other coordinates?Vector Decomposition using Standard Basis
Match each standard basis vector notation with its correct vector representation in a 4-dimensional space.
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Comparison of Complex and Euclidean Representations of RoPE Rotation
Consider a 4-dimensional real vector (\mathbf{x} = (1, 0, 0, 1)). A transformation is applied that groups the vector's components into consecutive pairs, treats each pair as a complex number (a + ib), and rotates them independently. The first pair (components 1 and 2) is rotated by an angle of (\pi/2) radians, and the second pair (components 3 and 4) is rotated by an angle of (\pi) radians. What is the resulting vector after this transformation?
Determining an Unknown Rotation Angle
A transformation is applied to a d-dimensional real vector by grouping its components into d/2 consecutive pairs. Each pair is treated as representing a point in a 2D plane and is independently rotated by a specific, non-zero angle. True or False: This transformation will always change the Euclidean norm (magnitude) of the original d-dimensional vector.