Question

Show that $\left(\text{3,1}\right),\left(-\text{2,}-\text{1}\right)$, and $\left(\text{4,3}\right)$ generate the additive group Z x Z.

Short answer

Answer

It is proved that (3,1) , (-2,-1), (4,3) generates the additive group Z x Z.

Step-by-step solution

Step by Step Solution Step 1: To find the points and (1,0) and (0,1)

The points are $\left(3,1\right),\left(-2,-1\right),\left(4,3\right)$

Consider (1,0) , which can be written as:

$$\begin{gathered} \left(1,0\right)=\left(3,1\right)+\left(-2,-1\right) \\ and, \\ \left(0,1\right)=\left(4,3\right)+2\left(-2,-1\right) \end{gathered}$$

By using this, all $\left(a,b\right)\in \mathbf{\text{Z}}\times \mathbf{\text{Z}}$ can be calculated.

To calculate (a,b)

Now (a,b) , can be written as:

$\begin{array}{c}\left(a,b\right)=a\left(1,0\right)+b\left(0,1\right)\\ =a\left(\left(3,1\right)+\left(-2,\text{\hspace{0.17em}}-1\right)\right)+b\left(\left(4,3\right)+2\left(-2,\text{\hspace{0.17em}}-1\right)\right)\\ =a\left(3,\text{\hspace{0.17em}}1\right)+\left(a+2b\right)\left(-2,-1\right)+b\left(4,3\right)\end{array}$

Hence, (3,1) , (-2,-1), (4,3) generates the additive groupZ x Z.