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Find $Ker(\phi)$ and $\phi(-3,2)$ for the given homomorphism $\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ where $\phi(1,0) = 3$ and $\phi(0,-1) = -5$

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I have no idea why how they came up with the mapping $\phi(m,n)$ themselves nor do I fully understand why the kernel is correct. My thinking for this problem was

$Ker(\phi) = \{(m,n) \in \mathbb{Z} \times \mathbb{Z}: \phi(m,n)=0\}$ so the problem now lies in finding $m$ and $n$.

What is the thinking behind the solution? I really don't understand how they come up with the function themselves

EDIT: okay I understand they are basically setting the function to 0 and that's how they came up with that kernel, but still no explanation on how they came up with the function

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    Thanks. That's probably where I found it, too.2012-11-19

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A typical element of ${\bf Z}\times{\bf Z}$ is $(a,b)$, which can be written as $(a,b)=a(1,0)+b(0,1)$ Since you are told what $\phi(1,0)$ is and you are told what $\phi(0,1)$ is, and you are told $\phi$ is a homomorphism, you can figure out what $\phi(a,b)$ is, and then you can figure out what $a,b$ have to do for $(a,b)$ to be in the kernel.

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    You can let $x=(1,0)$, and it is true that $\phi((1,0)-(0,1))=\phi(1,0)-\phi(0,1)$. But I'm not sure what you bring up this particular calculation.2012-11-19