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?
How is the transformation defined?
Originally Posted by tukeywilliams
yeah I think I got it now (didnt see how the transformation was defined).
How would you see if this is a linear transformation:
What exactly is that saying?
How is that a mapping from R^2 to R?
The customary definition of C[a,b] is the set of all continuous functions on [a,b].
What definition do you have to use?