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Is this a subspace of R3? The set of all vectors of the form where a+b+c = 0

Mathematics
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I think you misunderstood the question. a, b, and c are components of the vector, not vectors themselves. This set of vectors forms a plane through the origin, and therefore constitutes a subspace.
You can easily check the requirements for a subspace, also -- i.e. multiply any such vector by a scalar and its components will still add to zero, and the sum of any two such vectors is another vector whose components add to zero.
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