Math Help - coordinate system

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.

Originally Posted by MKLyon

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.

Draw the two pairs of cartesian (rectangular) axes having the same origin (0,0), such that (u,v) is rotated 45 degees from (x,y).

Mark a random point (x,0) on the positive x-axis. Draw the vertical line x = x, or project the (x,0) on the (u,v) axes.

What are the (u,v) coordinates of this line x = x?

u = x*sqrt(2)
v = -x*sqrt(2)
Or, x = (u/sqrt(2),0) +(0,-v/sqrt(2))
So,
x = u/sqrt(2) -v/sqrt(2)
x = (1/sqrt(2))*(u-v)
x = (1/2)sqrt(2) *(u-v) ---------------***

---------------------------------
For the y on the (u,v) axes.

Mark a random point (0,y) on the positive y-axis. Then project that onto the (u,v) axes.
u = y*sqrt(2)
v = y*sqrt(2)
Or, y = (u/sqrt(2),0) +(0,v/sqrt(2))
So, y = u/sqrt(2) +v/sqrt(2)
y = (1/2)sqrt(2) *(u+v) ---------------**