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I'm new to linear algebra, and I've stumbled upon a hypothesis that I'm not sure if it's right or not.

Assume $V$ is a group of vectors, where each of its vectors belong to $\mathbb R^4$. This means that each of the vectors components belongs to $\mathbb R$.

Based on this, can I assume that $V$ itself is a subgroup of $\mathbb R^4$ and therefore of $\mathbb R$? (I'm not necessarily asking about a subspace, just about a subgroup)

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Thank you!

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    Apparently I mean a group indeed, sorry for being unclear, I'm trying to translate the concepts into English.2012-11-13

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Well you can assume all sorts of things, but in this case assuming is not nearly as good as concluding something.

If you really mean it when you write "group of vectors," then you are probably assuming that adding two of them together results in a vector that is still in the group, and that the negative version of each vector is in the group, so that the addition operation is the vector operation we all know.

In that case, you would be right that the set $V$ is a subgroup of $R^4$. That's exactly the definition of a subgroup of a group: a subset that is closed under the operation of the containing group, and contains inverses to its own elements.

It is not necessarily a subspace, because there is the possibility it does not contain all scalar multiples of its elements.


Writing "$V$ is a subgroup of $R^4$ and therefore of $R$" does not make much sense, though, since $R^4$ is not a subset of $R$.

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    Thanks! this makes so much more sense now.2012-11-13