Assume x y in V so that 0 prove that 0 Suppose Vis a real f

Assume x, y in V so that <x,y> = 0, prove that <T(x), T(y)> =0

Suppose Vis a real finite dimensional inner product space and T is a linear operator on V such that ITfor all x in V. Prove that if x and y are orthogonal, then Tx) and T(y) are orthogonal

Solution

Assume x and y are orthogonal

so <x,y>=0..........(1)

(as T preserves norms).

||T(x+y)||² = ||x+y||² ...........................(2)

LHS = <Tx+Ty,Tx+Ty> = ||Tx||² +2<Tx,Ty> + ||Ty||²

                                    =||x||² +2<Tx,Ty> + ||y||² ...............(3)

RHS = <x+y,x+y> = ||x||² +2<x,y> + ||y||² ...........................(4)

Equating (3) and (4) , we obtain

<Tx,Ty>= <x,y>=0, as required:

x and y are orthogonal implies Tx and Ty are orthogonal