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Show that:

C=$\left\{\begin{pmatrix} a & b\\ -b & a \end{pmatrix}\bigg|a,b \in\mathbb{R}\right\}$

is a subring of R isomorphic to the field C of complex numbers.

where R is $\left\{\begin{pmatrix} a & b\\ c & d \end{pmatrix}\bigg|a,b,c,d \in\mathbb{R}\right\}$

I understand the concept of isomorphism but this question is a real stumbling block for me. Any help will be appreciated.

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The idea here is that the matrix $\begin{pmatrix}0&1\\-1&0\end{pmatrix} = J$ satisfies $J^2 = -I$, so it behaves like $i$ does. In light of this, define the map from $C$ to $\Bbb C$ to be \begin{align*} C&\to\Bbb C\\ \begin{pmatrix}a&b\\-b&a\end{pmatrix} = aI + bJ&\mapsto a + bi. \end{align*} It should be clear that this is surjective. Can you show it is injective? Can you show that it is a homomorphism of rings?

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