Math Is Fun Forum

basmah

Member

Registered: 2006-10-02

Posts: 18

i need help with this proof

Let V and W be two subspaces of a vector space U. Prove that the set

V + W = {u : u = v + w, where v € V and w € W}

is a subspace of U.

V = {(x,0) : x is a real number} and W = {(0,y) : y is a real number}.

** Bold = Vector and "€" = "such that"

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: i need help with this proof

The theorem that we need here is this:

If W is a set of one or more vectors from a vector space V, then W is a subspace of V iff:

Here we need to show that:

and:

For the first one:

Using the definitions of V and W:

(where s, t, x, y, are real numbers)

This addition simplifies to merely:

Now to show:

We let

(By the properties of real numbers)

Therefore:

So the first part is complete.

Now must show that:

which is pretty much the same process, except the truth of that depends on the fact that k(x, y) = (kx, ky) and by the properties of real numbers, (kx, ky) is an element of (V + W).

I hope I'm going about this the right way, this is sort of simple yet confusing.

basmah

Member

Registered: 2006-10-02

Posts: 18

Re: i need help with this proof

o yaa it kind of make sense now....thanks much!!!