Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I want to find a group homomorphism $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ which satisfies $f(1,0) = (2,6)$. Can any such homomorphism be made into an isomorphism?

share|improve this question
1  
A homomorphism of what? Groups? –  M.B. Oct 28 '12 at 23:03
    
Indeed. I will amend the statement at once. –  user44069 Oct 28 '12 at 23:13
add comment

1 Answer

up vote 8 down vote accepted

A homomorphism $\mathbb{Z}^2\rightarrow \mathbb{Z}^2$ can be represented as a $2\times 2$ matrix with integer coefficients with respect to the canonical bases. For example saying that $f(1,0)$ should be $(2,6)$ ammounts to specifying the first column of the matrix as $\left(\begin{smallmatrix}2\\6\end{smallmatrix}\right)$. The inverse of such a homomorphism, if it exists, will be given by the inverse matrix, since this is true of homomorphisms $\mathbb{Q}^n\rightarrow \mathbb{Q}^n$. With this information, you should be able to answer your own question.

share|improve this answer
    
Great hint! Thank you very much! –  user44069 Oct 28 '12 at 23:26
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.