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 = 0

Question

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?

Answer 1

Let $T:V\to W$. Then,

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