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Just a quick question. Is the following okay?:

The span of a set of vectors corresponds to a subspace inR^{n}.

But the span of a set of vectors can also be ALL of R^{n}, does that mean all ofR^{n}can be considered a subspace? Or does it mean the first definiton is not entirely correct, and instead a span of a set of vectors can simply be used to identify a subspace, if it exists.

I realise this is a fairly trivial question, but I would like to make sure that I haven't overlooked anything.

Here is an example of what I mean.

The span(v_{1},v_{2}), where [tex]v_{1} = \left(\begin{array}{cc}1\\0\end{array}\right), v_{2} = \left(\begin{array}{cc}0\\1\end{array}\right)[/tex] isR^{2}. It is also closed under addition and multiplication, and contains the zero vector, so it satisfies the requirements for a subspace; is it a subspace within, sayR^{n}with n>2? I don't think so, because the vector is composed of two components, and I was thinking a subspace would have to be within the confines of these two dimensions.

Any help appreciated, if it is not clear I can try to re-explain what I mean. I am hoping it's more of an issue of definition.

Thanks in advance.

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# Homework Help: Subspaces and spans

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