Why is it that W perp intersection with W is {0}?

Question

My question is:

If $$W^\perp$$ is a subspace of $$R^n$$

Then how is

$$W^\perp \cap W = [0] $$

Answer 1

Hint:

If you have a vector in both subspaces, call it $w$, then $w\cdot w=$ ...?

Answer 2

Suppose that $x\in W^\perp\cap W$. Then since $x$ is orthogonal to every vector in $W$, in particular, $x$ is orthogonal to itself i.e. $<x, x>=\lvert x\rvert^2=0$. So $x$ is the zero vector.

Also note that the zero subspace is a valid subspace of all vector spaces.

Answer 3

Given an inner-product space $X$ and some $W\subset X$, then, by definition,

$$W^\perp:=\left\{x\in X:\langle x,y\rangle=0\text{ for every }y\in W\right\}.$$

Use this and Cameron Williams's post above to answer your question.