Let \(v,w\) be two vectors of a vector space \(V\) over the field of complex numbers \(\mathbb C\) given in their column notations with respect to given finite basis:
\[ v=\pmatrix{ \alpha_{1}\cr \alpha_{2}\cr \vdots \cr \alpha_{n} \cr }, ~~~~~~~~w=\pmatrix{ \beta_{1}\cr \beta_{2}\cr \vdots \cr \beta_{n} \cr }, \] the dot product, (or inner product, or scalar product) of \(v\) and \(w\) is the function complex \(\langle v,w\rangle:V\times V\mapsto\mathbb C\)
\[\langle v,w\rangle:=\pmatrix{ \alpha_{1}~ \alpha_{2}~ \dots ~ \alpha_{n} }^*\cdot\pmatrix{ \beta_{1}\cr \beta_{2}\cr \vdots \cr \beta_{n} \cr }=\alpha_1^*\beta_1+\alpha_2^*\beta_2+\ldots+\alpha_n^*\beta_n\in \mathbb C,\]
where \(\alpha_i^*\) denotes the complex conjugate of \(\alpha_i\) for all \(i=1,\ldots,n\).
