
Linear algebra with transformation.
Posted:
Feb 5, 2013 4:08 AM


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).
 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 ?

