# Definition: Dot Product, Inner Product, Scalar Product (Complex Case)

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 $$i=1,\ldots,n$$.

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Wille, D; Holz, M: "Repetitorium der Linearen Algebra", Binomi Verlag, 1994