Condensed Notation for Pairing Real Vector Elements into Complex Numbers
A transformation that reinterprets a real-valued vector as a complex-valued vector by pairing adjacent elements can be expressed in a highly condensed notation. The source text presents this as x = x x ... x = x+ixx+ix...x+ix. This notation equates the original real-valued vector x (represented as a sequence x x ... x) with a new sequence of complex numbers. Each term x+ix in the sequence represents a complex number formed by pairing two consecutive elements from the original real vector, such as x_1 + ix_2, x_3 + ix_4, and so on.
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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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Condensed Notation for Pairing Real Vector Elements into Complex Numbers
A real-valued vector of even dimension can be reinterpreted as a complex-valued vector of half the dimension by pairing adjacent elements. The first element of each pair forms the real part and the second element forms the imaginary part of a new complex number. For instance, the pair of real numbers
(a, b)would become the single complex numbera + ib. Given the real-valued vectorv = (8, -2, 0, 5, 1, 3), which of the following represents the correct transformation into a complex-valued vector?Real to Complex Vector Transformation
A transformation process creates a complex-valued vector by pairing adjacent elements of a real-valued vector of even dimension. The first element of each pair becomes the real part and the second becomes the imaginary part of a new complex number (e.g., the pair
(a, b)becomesa + ib). If this process results in the complex-valued vectorc = (4 + 2i, -1 + 5i, 0 - 8i), what was the original real-valued vector?
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A transformation is described using the condensed notation
x = x x ... x = x+ixx+ix...x+ix, where a real-valued vector is reinterpreted as a sequence of complex numbers by pairing adjacent elements. Given the real-valued vectorv = [2, 5, -1, 8, 0, 3], which of the following complex-valued vectors is the correct result of applying this transformation?Reconstructing a Real Vector from its Complex Representation
Analysis of a Vector Transformation Error