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My problem: Show the linear dependence on chosen set of scalars. Find the subset $M\subset \mathbb C^2$ which is linear independent if we consider $\mathbb C^2$ as a vector space over $\mathbb C$ and is linear dependent if we consider $\mathbb C^2$ as a vector space over $\mathbb R$.

Well, that given set is dependent if there is a nontrivial linear combination which is zero, otherwise independent.

Unfortunately I haven't been succesful in that searching. Any ideas please? Thanks.

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    Don´t You mean the given set is dependent if there is a nontrivial linear combination which is zero and otherwise independent? – 2017-02-08
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    Think of $(1,0),(i,0),(0,1),(0,i)$... – 2017-02-08
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    Yes, I meant that, I will edit it. – 2017-02-08
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    the trick is to use the isomorphism $\mathbb{C}^2\cong \mathbb{R}^4$ as $\mathbb{R}$-vectorspaces – 2017-02-08
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    I see, thank you very much! I haven't realised that. – 2017-02-08
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    You´re welcome! – 2017-02-08

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