Question

let f be an automorphism of R^2 with
f((1)) = ( 2 ) and f((1)) = (0)
(3) (-1) (4) (1)

Find f( ( 0) )
(-1)

Answer 1

Let \[v_1=(1,3),v_2=(1,4)\]if it was possible to write (0,-1) as a linear combination of \[v_1\] and \[v_2\], then you could use the morphism property:\[f(0,-1)=f(c_1v_1+c_2v_2)=c_1f(v_1)+c_2f(v_2)\] So basically you want to rewrite (0,-1) in terms of the two vectors given.

Answer 2

so what woud c1 and c2 be? and does this mean the answer wouldnt be a vector?

Answer 3

To continue off of joe's work, notice that \[\left( \begin{matrix} 1\\3\end{matrix}\right)-\left( \begin{matrix} 1\\4\end{matrix}\right)=\left( \begin{matrix} 0\\-1\end{matrix}\right)\]Since automorphisms are homomorphisms, \[f\left( \begin{matrix} 0\\-1\end{matrix}\right)= f\left(\left( \begin{matrix} 1\\3\end{matrix}\right)-\left( \begin{matrix} 1\\4\end{matrix}\right)\right)\]From there you should be able to easily solve.

Answer 4

ohhh. okay. big help to see it. thank you.

Answer 5

You're welcome.