Date: Feb 5, 2013 4:08 AM
Author: mina_world
Subject: Linear algebra with transformation.

Hello teacher~

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).

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Let any line be y=ax+b. (x, ax+b)

Im f = <f(1,0), f(0,1)>
f(1,0) and f(0,1) are not linearly independent.
so, Im f = <f(1,0)> or <f(0,1)>
so, Im f is a line passing through the origin.

Hm, how do you show that "fixed point" part ?
Of course, I know a proper example.
ex)
let be a line x+y+1=0
f(x,y) = (x+y, x+y)
A =
(1 1)
(1 1)
so, Rank(A) = 1
Then, f(x,y) = (x+y, x+y) = (-1, -1). It's really a fixed point.

If you does not use these examples, how do you prove it ?