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A standard basis vector, often denoted as $\vec{e}_k$, is a vector within a given vector space that has a value of 1 in its k-th coordinate and 0 in all other coordinates. These vectors form a basis, meaning any vector in the space can be represented as a linear combination of them.

Standard Basis Vector

The transformation $C(\mathbf{x}, t\theta)$ performs component-wise rotations on a vector in complex space. It is defined by the formula: $$C(\mathbf{x}, t\theta) = \sum_{k=1}^{d/2} x'_{k} e^{it\theta_k} \vec{e}_k$$ This equation represents the final rotated vector as a linear combination. Each term in the sum consists of a complex component $x'_{k}$ (derived from the original real vector $\mathbf{x}$) rotated by an angle $t\theta_k$, and then scaled by its corresponding standard basis vector $\vec{e}_k$. The term $\vec{e}_k$ represents a vector with a value of 1 in the k-th position and zeros elsewhere.

Component-wise Vector Rotation in Complex Space

Consider a vector `v` in a 3-dimensional space, defined as `v = [4, -7, 2]`. Which of the following options correctly expresses `v` as 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?

Consider a vector $\vec{v}$ in a 4-dimensional space, represented as $\vec{v} = [c_1, c_2, c_3, c_4]$. Explain how this vector can be expressed as a weighted sum of four other vectors, where each of these four vectors has a value of 1 in a single position and 0s in all other positions. In your explanation, explicitly write out these four special vectors and the final equation for $\vec{v}$.

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