Confused about the proof of U = U + U, where U is a subspace

[PhillipKP]

Homework Statement

Hi I need help understanding a proof. This is my first time in a pure math class, so proofs of this type are a little weird to me.

If U is a subspace of the vectorspace V, what is U+U?

Homework Equations

The proof:


$$(v_{1}+v_{2})\in U+U$$

As $$v_{1},v_{2}\in U$$and $$U$$ is a subspace of $$V$$,

$$(v_{1}+v_{2})\in U$$

Thus $$U+U\subseteq U$$

Now let
$$v\in U$$.

Then as $$0\in U$$,

$$v=(v+0)\in U+U.$$

Thus $$U\subseteq U+U$$

$$\therefore U=U+U$$

The Attempt at a Solution

I don't understand why the first part proves $$U+U\subseteq U$$ rather than $$U\subseteq U+U$$.

Similarly, I don't understand why the second part proves $$U\subseteq U+U$$ rather than proving $$U+U\subseteq U$$.

I guess I just don't understand why each part proves 1 direction of the equality by not the other direction.


Thanks for any help you can provide.

 

[Dick]

I don't understand what's confusing you. In the first part you picked a general element of U+U and showed that it's in U, right? If I pick any element of A and show that it is in B, that means A is a subset of B, also right?

 

[PhillipKP]

Ok I agree with you. But why is the second part of the proof so much different from the first part?

Couldn't I just exchange the 1st and 3rd line of the proof to prove the other direction?

 

[Dick]

The second part is different, uh, because it's different. Now you want to take an element of U and show it's in U+U. The first part uses the fact that vector spaces are closed under addition. The second part uses that they have an additive identity. They really are different.

 

[VietDao29]

To prove that 2 set A and B are equal, one has to prove that A is a subset of B, and B is also a subset of A. Or written formally:

$$A = B \Leftrightarrow \left\{ \begin{array}{ccc} A & \subset & B \\ B & \subset & A \end{array} \right.$$

Or in words, A are B are equal, iff every element of A is contained in B, and vice-versa.

Vector-space (or sub-space) can be understood as a set of vectors with 2 predefined operators: vector addition, and scalar multiplication.

So to prove 2 subspace A, and B are equal, one just needs to prove:

$$A = B \Leftrightarrow \left\{ \begin{array}{ccc} A & \subset & B \\ B & \subset & A \end{array} \right.$$

Homework Statement

Hi I need help understanding a proof. This is my first time in a pure math class, so proofs of this type are a little weird to me.

If U is a subspace of the vectorspace V, what is U+U?

Homework Equations

The proof:$$(v_{1}+v_{2})\in U+U$$

We'll now be proving that every element of U + U is in U. First, take out a random vector in U + U, namely v₁ + v₂.

As $$v_{1},v_{2}\in U$$and $$U$$ is a subspace of $$V$$,

$$(v_{1}+v_{2})\in U$$

We have proved that this vector is also contained in U. And hence, we have:

Thus $$U+U\subseteq U$$

Our second step is to prove that every element of U is contained in U + U.

Now let
$$v\in U$$.

Consider a random vector v in U.

Then as $$0\in U$$,

$$v=(v+0)\in U+U.$$

Thus $$U\subseteq U+U$$

Our second goal is here.. And hence:

$$\therefore U=U+U$$

The Attempt at a Solution

I don't understand why the first part proves $$U+U\subseteq U$$ rather than $$U\subseteq U+U$$.

Similarly, I don't understand why the second part proves $$U\subseteq U+U$$ rather than proving $$U+U\subseteq U$$.

I guess I just don't understand why each part proves 1 direction of the equality by not the other direction.Thanks for any help you can provide.

Are there any more steps that confuse you?