# Inner product space csir 2016

Defined that $$\langle x,y \rangle=\langle Ax,Ay\rangle$$

Prove that above given define a inner product iff A is invertible.

I know that $$\langle Ax,Ay\rangle=xA^{t}Ay$$ and for inner product we have to show that $$\langle Ax,Ay \rangle > 0$$ but further i don't know how to proceed

Please help!

## 1 Answer

You should use different symbol for the inner product on the left. I will use a prime. $$\langle Ax, Ax \rangle=0$$ iff $$\|Ax\|=0$$ iff $$Ax=0$$. So the condition that $$\langle x, x \rangle '=0$$ implies $$x=0$$ is satisfied only when $$A$$ satisfies the property that $$Ax=0$$ implies $$x=0$$ which means $$A$$ is invertible. The other properties of inner product are straightforward.

• sir,if $Ax=0$ implies $x=0$ then how can we say A is invertible? – gaurav saini May 10 at 9:04
• If $Ax=0$ implies $x=0$ then $A$ is one-to-one, hence invertible. I am assuming that you are working in a finite dimensional vector space. – Kavi Rama Murthy May 10 at 9:12
• sir, $Ax=0$ implies $x=0$ so the kernal of $A={0}$ which implies A is one -one and in finite dimension space if A is one -one then it becomes invertivle.. i got it Thnakyou sir. – gaurav saini May 10 at 9:22