Understanding Projection: Clarifying Confusion in Orthogonal Projections

[holezch]

Homework Statement

Hi, I got tied up with something..
I have a question that says if a projection P satisfies || P v || <= || v ||
then P is an orthogonal projection.. but if I drew in |R^2, a x-axis and a y=x line, and projected some vector onto the y = x line.. I still get || Pv || <= || V || ? I think Iam doing something wrong

thanks

 

[LCKurtz]

When you say you project V onto the y = x line, are you projecting it perpendicular to the line? I mean thinking of the tip of V projecting perpendicularly to the line? And if so, what's the problem? If you are projecting, for example, vertically to the line then you don't have your norm inequality.