Date: Feb 5, 2013 5:57 AM
Author: Jose Carlos Santos
Subject: Re: Linear algebra with transformation.

On 05/02/2013 09:08, mina_world wrote:

> 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).

Best regards,

Jose Carlos Santos