Let \(v,w\) be two vectors of a finitely dimensional vector space \(V\) over a the field \(F\) 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 inner product, (or dot product, or scalar product) of \(v\) and \(w\) is a function \(\langle v,w\rangle:V\times V\mapsto F\) defined by \[\langle v,w\rangle:=\pmatrix{ \alpha_{1}~ \alpha_{2}~ \dots ~ \alpha_{n} }\cdot\pmatrix{ \beta_{1}\cr \beta_{2}\cr \vdots \cr \beta_{n} \cr }=\alpha_1\beta_1+\alpha_2\beta_2+\ldots+\alpha_n\beta_n\in F.\]
Corollaries: 1
Corollaries: 1
Definitions: 2 3 4 5
Proofs: 6
Propositions: 7 8
