How would you prove this theorem?

Question

Theorem: Let S = (v1, v2,..... vk) be a set of vectors in a vector space V Then span(S) is a subspace of V

using the hint: (Start with two arbitrary elements in span(S) and show that span(S) is closed
under addition and scalar multiplication.)

Answer 1

HI: I tried to answer your email but it said that your email address has not been confirmed and so the email didn't go through.

alpha and beta are scalers for the vector space V. This problem is actually so easy that people don't believe what they've done. You really just need to show that Span S behaves like V.

Span S must be a vector space itself. Two elements from the span is
u=a1*v1 + a2*v2...ak*vk
and
w=b1*v1 + b2*v2...bk*vk

show that alpha*u+beta*w is in Span S where alpha, beta are scalers for your vector space.

alpha*u+beta*w = (alpha*a1+beta*b1)*v1 + ....
is a linear combination of the vectors in Span S, thus it's in Span S.

So, Span S is not empty and is closed, thus its a subspace.

Answer 2

Uh... I would start by showing that span(S) is closed under addition and scalar multiplication. This one really is as simple as following the instructions.

Edit: In point of fact, I DO know how to show that span(S) is closed under addition and scalar multiplication. Any vector in span(S) is a linear combination of the vectors in S - i.e. c_1v_1 + c_2v_2 +... c_nv_n. Add two such vectors together and you get (c_1+d_1)v_1 + (c_2+d_2)v_2... + (c_n+d_n)v_n, which is clearly in span(S). Similarly, multiplying by the scalar k you get kc_1v_1 + kc_2v_2... kc_nv_n. Of course, the previous poster told you all this already. The fact is that this IS so bloody simple that typing it out is really the hardest part of the problem. Why you choose to assume that I don't KNOW the answer, simply because I have chosen not to TYPE OUT the answer, especially when the answer has already been adequately given, is more than I can see.

Answer 3

you need to do the following
-show that the 0 vector is an element of Span(S),
-show that for any two vector in Span(S), their sum is also in Span(S)