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 ?