Let x ∈ Rn and let p be the orthogonal projection of x onto W where W is a subspace of Rn. Provethat for all y ∈ W ,
||x−(p+y)||2 =||x−p||2 +||y||2.
Now expanding out using the defn of norm (dot product) doesn't get me very far, any hints?
Let x ∈ Rn and let p be the orthogonal projection of x onto W where W is a subspace of Rn. Provethat for all y ∈ W ,
||x−(p+y)||2 =||x−p||2 +||y||2.
Now expanding out using the defn of norm (dot product) doesn't get me very far, any hints?
$\displaystyle \|(x-p)+y)\|^2= <(x-p)+y,(x-p)+y>\text{ } =$
$\displaystyle <x-p,x-p> -2<x-p,y>+<y,y> \text{ }=$
$\displaystyle \|x-p\|^2+\|y\|^2-2<x-p,y>$
now y is in W, x-p is orthogonal to W so
$\displaystyle <x-p,y>=0$