Gregory's question at Yahoo Answers (Inner product space)

[Fernando Revilla]

Here is the question:

Prove that T is linear: T(x)=<x,y>z for all x in V.

Here is a link to the question:

Linear algebra- prove linearity of an inner product space? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

 

[Fernando Revilla]

Hello Gregory,

If $V$ is the given vector space over the field $\mathbb{F}\;(\mathbb{R}$ or $\mathbb{C}$), for all $\lambda,\mu\in\mathbb{F}$ scalars, for all $u,v\in V$ vectors and using the axioms of inner product: $$\begin{aligned}T(\lambda u+\mu v)&=<\lambda u+\mu v,y>z\\&=(\lambda<u,y>+\mu<v,y>)z\\&=\lambda<u,y>z+\mu<v,y>z\\&=\lambda T(u)+\mu T(v)\\&\Rightarrow T\mbox{ is linear}\end{aligned}$$