Prove that x is in W if and only if $proj_W$(x) = x

Question

Let W be a subspace of $R^n$, and let x be a vector in a $R^n$. I need to prove that x is in W if and only if $proj_W$(x) = x

Can I have some hints on how to get started with this problem?

Answer 1

This is trivial once you prove that

1. $proj_W (\mathbb{R}^n) = W$

2. $proj_W|_W = id_W$.

These two properties are all the ingredients you need, and they are straightforward to verify just from the definition of $proj_W$.

Now, for all $x \in W$, $x = proj_W(x)$. Viceversa, if $x = proj_W(x)$, then $x \in proj_W (\mathbb{R}^n) = W$.