Dot Product

Let u , v : R n : 𝑢 𝑣 superscript 𝑅 𝑛 u,v:R^{n} be a pair of vectors with entries in some Ring R 𝑅 R . The dot product u ⋅ v : R n : ⋅ 𝑢 𝑣 superscript 𝑅 𝑛 u\cdot v:R^{n} is given by

u⋅v=∑i=0n−1ui⁢vi⋅𝑢𝑣superscriptsubscript𝑖0𝑛1subscript𝑢𝑖subscript𝑣𝑖u\cdot v=\sum_{i=0}^{n-1}u_{i}v_{i}

Alternatively, we can express the dot product as u 𝖳 ⁢ v superscript 𝑢 𝖳 𝑣 u^{\mathsf{T}}v , treating u 𝑢 u as a Covector.