Representing 2D Vector Rotation in Complex Space
A common method for conceptualizing vector rotation involves mapping vectors from a two-dimensional Euclidean space () to the complex plane (). This is achieved through a bijective linear transformation, which allows a 2D vector to be represented as a single complex number.
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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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Rotation as Complex Number Multiplication
A two-dimensional vector, such as
v = [a, b], can be represented as a single number in the complex plane. This is done by mapping the first component of the vector to the real part of the complex number and the second component to the imaginary part. Given this transformation, which of the following complex numbers correctly represents the vectorx = [5, -2]?When mapping two-dimensional vectors to the complex plane, it is possible for two distinct vectors to be represented by the exact same complex number.
A common method for analyzing vector rotation involves representing two-dimensional vectors as complex numbers. Match each 2D vector on the left with its corresponding representation in the complex plane on the right.