# Let T be a projection on a finite dimensional inner product space V. Consider x \in R(T) and y \in R(T)\bot . Is the inner product <x,y> = 0

Suppose T is a projection on a finite dimensional inner product space V. Consider $$x\in R(T) ,y\in R(T)^{\bot}$$ Does the inner product < x,y> = 0? $$\forall x\in R(T),\forall y\in R(T)^{\bot}$$ I understand that in a subspace if $$a\in W, b\in W^{\bot}$$ then the inner product < a,b> = 0.

If < x,y> = 0 does not hold for a projection, does it hold for orthogonal projections?

• $R(T)$ is a subspace, so you can use that fact directly... Apr 11 '20 at 4:15

Let $$T:V\to W$$. Then,
Yes $$=0 \;\forall x\in R(T) ,y\in R(T)^{\bot}$$. Since $$R(T)$$ is subspace of W, and the definition of $$R(T)^{\bot}=\{y\in W|v\cdot w=0\; \forall w \in R(T) \}$$