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samuel12

Member

Registered: 2010-07-17

Posts: 19

vector spaces and subspaces

This is Theorem 5.2

Let V be a vector space and let W be a nonempty subset of V. Then w is a subspace of V if and only if the following conditions hold:

a. If u and v are in W, then u + v is in W.
b. If u is in W and c is a scalar, then cu is in W.

Now for the question:

In exercise 25, use Theorem 5.2 to determine whether W is a subspace of V.

Can someone please explain to me what is going on here, thanks   (if it makes a difference the column vector above is also ment to have squigly brackets around it, but i couldnt do that AND the parenthesis.)

samuel12

Member

Registered: 2010-07-17

Posts: 19

Re: vector spaces and subspaces

Nevermind i think i've got it, I'll post my solution tomorrow.

samuel12

Member

Registered: 2010-07-17

Posts: 19

Re: vector spaces and subspaces

Solution:

is a nonempty set because it contins the zero vector 0.

(i.e. Let)

Let u  and  v  be in 

Say,

and

Then

So u+v is also in

  (because it has the right form)

Siilarly if k is a scalar, then

So ku is in

Thus,

is a nonempty set of

that is closed under addition and scalar multiplication. Therefore,

is a subspace of

, by Theorem 5.2

Last edited by samuel12 (2010-07-22 08:09:31)