Subspace & Span in Rn: Definition & Examples

[nobahar]

Hello!
Just a quick question. Is the following okay?:
The span of a set of vectors corresponds to a subspace in Rⁿ.
But the span of a set of vectors can also be ALL of Rⁿ, does that mean all of Rⁿ 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 realize 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(v₁,v₂), where $$v_{1} = \left(\begin{array}{cc}1\\0\end{array}\right), v_{2} = \left(\begin{array}{cc}0\\1\end{array}\right)$$ is R². 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, say Rⁿ 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.

 

[vela]

Hello!
Just a quick question. Is the following okay?:
The span of a set of vectors corresponds to a subspace in Rⁿ.
But the span of a set of vectors can also be ALL of Rⁿ, does that mean all of Rⁿ can be considered a subspace?

Yes, a vector space is a subspace of itself.

 

[rasmhop]

$\mathbb{R}^n$ is a subspace of itself (you can verify with the definition that it satisfies the properties of a subspace).

 

[nobahar]

Thankyou Vela and Rasmhop.