Mathematics, 10.07.2019 03:30 JANA279
Let g' be the group of real matricies of the form [1 x 0 1]. is the map that sends x to this matrix an isomorphism?
Answers: 1
. So, each matrix is identified by its top-right coefficient, since the other three entries remain constant.![\phi:\ \mathbb{R} \to G,\quad \phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right]](/tpl/images/0071/8322/2b08c.png)
. Then, you trivially have 
, because they are two different matrices:![\phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right],\quad \phi(y) = \left[\begin{array}{cc}1&y\\0&1\end{array}\right]](/tpl/images/0071/8322/9da28.png)
![\phi(x) = \left[\begin{array}{cc}1&x\\0&1\end{array}\right]](/tpl/images/0071/8322/d7c87.png)
and its inverse are both homomorphisms: if we consider the usual product between matrices to be the operation for the group G and the real numbers to be an additive group, we have![\phi (x+y) = \left[\begin{array}{cc}1&x+y\\0&1\end{array}\right] = \left[\begin{array}{cc}1&x\\0&1\end{array}\right] \cdot \left[\begin{array}{cc}1&y\\0&1\end{array}\right] = \phi(x) \cdot \phi(y)](/tpl/images/0071/8322/d46f6.png)
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Let g' be the group of real matricies of the form [1 x 0 1]. is the map that sends x to this matrix...
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