linear algebra: orthogonal projection?

Let x ϵ Rn and let p be the orthogonal projection of x onto W where W is a subspace of Rn. Prove that for all yϵW, ||x-(p+y)||^2 = ||x-p||^2 + ||y||^2. It may help to draw a picture and interpret the result geometrically.

I don't get how you would draw the picture since I don't know how I would draw the y part. Can someone please help me?

Question

linear algebra: orthogonal projection?

Let x ϵ Rn and let p be the orthogonal projection of x onto W where W is a subspace of Rn. Prove that for all yϵW, ||x-(p+y)||^2 = ||x-p||^2 + ||y||^2. It may help to draw a picture and interpret the result geometrically. 

I don't get how you would draw the picture since I don't know how I would draw the y part. Can someone please help me?

phi · Answer

I would draw a picture for R^2
|dw:1339706906812:dw|

he66666 · Answer

I don't get this picture :/ how do you draw the y?

amistre64 · Answer

im not sure about the information in the question either; but an orthoPro can be thought of as a shadow on the ground.

amistre64 · Answer

the picture drawn, I would translate as, the shadow of the top arrow on the bottom arrow is the projection of the top onto the bottom

amistre64 · Answer

the rest of it looks like a circle equation to me .... not that that helps

phi · Answer

|dw:1339767463980:dw|
the picture shows x in R^2 projected onto W  ( a vector).  It shows x can be decomposed into p  that lies parallel to W, and z that lies orthogonal (perpendicular) to W

vector addition shows x= p+z  or z= x-p