what does it mean "A function that takes two vectors and return scalar?

Question

what does it mean?

A function that takes two vectors and return real number is real inner product space

and complex product space if it returns a complex number

Can anyone give me an example for that?

Answer 1

Consider the vector space $\mathbb{R}^2$. Given two vectors $v = (a,b)$ and $w = (c,d)$, where $a,b,c,d$ are real numbers, define $f(v,w) = f((a,b), (c,d)) = ac + bd$. This is the usual dot product (inner product) associated wtih $\mathbb{R}^2$.

For a complex example, consider $\mathbb{C}^2$. Given two vectors $v = (a,b)$ and $w = (c,d)$, where $a,b,c,d$ are complex numbers, define $f(v,w) = f((a,b), (c,d)) = a\overline{c} + b\overline{d}$, where the bar denotes complex conjugation. This is the usual dot product (inner product) associated wtih $\mathbb{C}^2$.

Answer 2

dot product usually scalar projection of one vector onto another. Dot product is just a specific version of inner product. You can define an inner product whose formula looks a bit different than dot product... for example, consider two vectors $$x_1$$ and $$x_2$$ and then you can arbitrarily define the inner product like $$<x_1,x_2>=x_1 x_2+2x_1 x_2+x_2 x_1(*)$$

in linear algebra course, when applying the Gram–Schmidt algorithm most of the questions define the inner product different than dot product...something like (*)

Answer 3

If you meant that what you said is the definition of the inner product then it's wrong. Let $E$ a vector space over $\Bbb R$. A map $$\varphi: E\times E\to \Bbb R,\; (u,v)\mapsto \varphi(u,v)$$ is an inner product if $\varphi$ is bilinear symmetric definite positive which means

• $\varphi$ is linear relative on each variable: (bilinear)
• $\forall (u,v)\in E^2,\; \varphi(u,v)=\varphi(v,u)$ :(symmetric)
• $\varphi(u,u)\ge 0\;\forall u\in E$ and $\varphi(u,u)=0\implies u=0$:(definite positive).