> f : R^2 -> R^2 be a linear transformation. > A is the standard matrix of f. > Rank(A) = 1 > > Then, > f transforms any line into (a line passing through the origin OR a fixed > point).
Proof: Let _v_ be a non-null vector from the image of _f_; let _w_ be a non-null element from the kernel of _f_ (it must exist; otherwise, _f_ would have rank 2). Now, take an arbitrary line. If it is a line of the form u + Rw for some vector _u_, then its image is just f(u). Otherwise, its image is the image of _f_, which is Rv (which is a line passing through the origin).