Let (u,v) be a coordinate system whose origin is the same as the one for the (x,y) system and which is obtained by rotating the (x,y) coordinate axes by 45 degrees. How would I prove these coordinate transformation formulas

x = (sqrt(2)/2)*(u-v) and y = (sqrt(2)/2)*(u+v)?

I would rather use geometric reasoning than linear algebra. I'm sure it has something to do with sin(45) or cos(45), but I just can't see how to transform it.