Linear Algebra showing a subspace

[ibkev]

I'm having trouble getting started on this one and I'd really appreciate some hints. This question comes from Macdonald's Linear and Geometric Algebra book that I'm using for self study, problem 2.2.4.

Homework Statement

Let U₁ and U₂ be subspaces of a vector space V.
Let U be the set of all vectors of the form u₁ + u₂, where u₁ is a member of U₁ and u₂ is a member of U₂.
Show that U is a subspace of V.

Homework Equations

U inherits it's operations from V (through U₁ and U₂) and so we have to show that scalar.mult and vec.add are closed for U.
a(bv)=(ab)v
av + bv = (a+b)v

The Attempt at a Solution

Since U₁ and U₂ are a subset of V, then adding any u₁ to u₂ may not be closed wrt either U₁ or U₂ but I think they should be closed wrt V. Intuitively that seems true but I get stuck trying to be more concrete than that.

 

[vela]

It often helps to write down explicitly what you're trying to show. For example, one of the things you want to show is if $\vec{x},\vec{y} \in U$, then $\vec{x}+\vec{y}\in U$.

When you say $\vec{x} \in U$, it means there are vectors $\vec{x}_1 \in U_1$ and $\vec{x}_2 \in U_2$, such that $\vec{x} = \vec{x}_1 + \vec{x}_2$. And so on…

 

[PeroK]

I'm having trouble getting started on this one and I'd really appreciate some hints.

The Attempt at a Solution

Since U₁ and U₂ are a subset of V, then adding any u₁ to u₂ may not be closed wrt either U₁ or U₂ but I think they should be closed wrt V. Intuitively that seems true but I get stuck trying to be more concrete than that.

Can you say what it means for $U$ to be a subspace of $V$?

Hint: Let $u, v \in U$ and $a$ be a scalar ...

And @vela has given you even more of a helping hand!

 

[member 587159]

We define:

a and b vectors, W1 and W2 subsets of a vectorspace V.

Definition: W1 + W2 = {a+b|a ∈ W1 and b ∈ W2}

Choose 2 vectors a,b element of W1. Choose 2 vectors c,d elements of W2. W1 and W2 are subsets of a vectorspace V.
Then $$a + c ∈ (W1 + W2)$$ and $$b + d ∈ (W1 + W2)$$. Try to continue.

You need to proof that (a+b) + (c+d) ∈ (W1 + W2), this means: a + b ∈ W1 and c + d ∈ W2

 

[PeroK]

We define:

a and b vectors, W1 and W2 subsets of a vectorspace V.

Definition: W1 + W2 = {a+b|a ∈ W1 and b ∈ W2}

Choose 2 vectors a,b element of W1. Choose 2 vectors c,d elements of W2. W1 and W2 are subsets of a vectorspace V.
Then $$a + c ∈ (W1 + W2)$$ and $$b + d ∈ (W1 + W2)$$. Try to continue.

You need to proof that (a+b) + (c+d) ∈ (W1 + W2), this means: a + b ∈ W1 and c + d ∈ W2

That's giving too much away. You need to let the OP do the problem himself.

 

[ibkev]

What an amazing community.
Thanks everyone for the replies! I'll take it from here

 

[Mark44]

That's giving too much away. You need to let the OP do the problem himself.

So noted, and addressed. I would have deleted the offending post, but the OP has already seen it.