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Re: Linear algebra with transformation.
Posted:
Feb 5, 2013 10:47 PM


"José Carlos Santos" ?? ??? ??? ??????. anc6ruF2kplU1@mid.individual.net...
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 nonnull vector from the image of _f_; let _w_ be a nonnull 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).
Oh, thank you very much for your detailed proof.



