OK. I know that to verify a linear transformation you must show that L(u + v) = L(u) + L(v) and L(ku) = kL(u). for every u and v in R.

I guess I don't quite understand this. Could someone show me how to verify L(x, y, z) = (x+y, 0, 2x-z)...

I want to say u = (x1, y1, z1) and v=(x2, y2, z2). Right. How in the world do it make it look resemble what's given if I L( u + v) = L( x1 + x2, y1 + y2, z1 + z2) = (x1, y1, z1) + (x2, y2, z2) = L(u) + L(v).

I was given some help, and supposedly the answer resembles:

L( x1 + x2, y1 + y2, z1 + z2) = (x1 + x2+y1 + y2,0,2*(x1 + x2) - (z1 + z2)) = L(u) + L(v)

You can just arbitrarily move things around to make it look like what you want??

How would anything NOT be a transformation? Are there rules to follow?

If this is how I verify the example I gave above...then when something NOT be a linear transformation. PLEASE HELP. I'm so confused...