I am trying to show that the R^2 --> R is NOT a linear transformation

So I let (x,y) be a vector in R^2. Then the transformation is:

T(x,y) = x

T(c(x,y)) = T((cx,cy)) = cx = cT(x,y)

Let (x1, y1) and (x2, y2) be vectors in R^2. Then T((x1,y1) + (x2,y2)) = T(x1+x2, y1+y2) = x1 + x2 = T(x1,y1) + T(x2,y2). But this should be false. What am I doing wrong?

The same for R^2 --> R^2 (not a linear transformation)

I let (x,y) be in R^2 and c (a scalar) be in R. Then:

T(x,y) = (x,y)

T(c(x,y)) = T(cx,cy) = cT(x,y). But this should be false according to the book. Should I even be choosing vectors at all?

The last step is what I got, which should conclude that it is not a linear transformation. However, clearly T(u + v) = T(u) + T(v). Did I choose my vectors wrong?