Is Span{u1,u2,...,um} a Subspace of R^n?

[squenshl]

I have a problem.
Suppose that {u1,u1,...,um} are vectors in R^n. Prove, dircetly that span{u1,u2,...,um} is a subspace of R^n.
How would I go by doing this?

 

[Hurkyl]

Well, directly, as the question asks. Where are you stuck?

 

[adriank]

Depends on your definition of span (my favourite being span {u₁, ..., u_m} = the smallest subspace containing u₁, ..., u_m, from which the result is trivial). ;)

You probably define span {u₁, ..., u_m} = {a₁u₁ + ... + a_mu_m | a₁, ..., a_m in R}. Just apply your definition of (or test for) a subspace.

 

[squenshl]

I just need to know how to get started.

 

[Cantab Morgan]

I just need to know how to get started.

Well, your span (probably meaning as adriank pointed out) is the the set of all linear combinations of those vectors. So that's clearly a subset of your vector space, right?

So, what's the difference between a subspace of a vector space and just a plain old subset? What's the magic property that spaces have that sets don't? Then you just need to demonstrate that your subset has it.