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?

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- Dec 12th 2013, 07:36 PMturbozzOrthogonal Projections Proof
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? - Dec 12th 2013, 08:36 PMromsekRe: Orthogonal Projections Proof
$\displaystyle \|(x-p)+y)\|^2= <(x-p)+y,(x-p)+y>\text{ } =$

$\displaystyle <x-p,x-p> -2<x-p,y>+<y,y> \text{ }=$

$\displaystyle \|x-p\|^2+\|y\|^2-2<x-p,y>$

now y is in W, x-p is orthogonal to W so

$\displaystyle <x-p,y>=0$ - Dec 12th 2013, 10:12 PMromsekRe: Orthogonal Projections Proof
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- Dec 12th 2013, 10:18 PMromsekRe: Orthogonal Projections Proof