Linear Algebra showing a subspace

In summary, a subspace in linear algebra is a subset of a vector space that satisfies all the properties of a vector space. To determine if a subset is a subspace, it must satisfy closure under addition, closure under scalar multiplication, and contain the zero vector. The study of subspaces is important because it allows for a better understanding of complex vector spaces. A subspace can span the entire vector space, but not always. It is related to the concept of linear independence, as it is made up of linearly independent vectors.
  • #1
ibkev
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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.
 
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  • #2
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…
 
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  • #3
ibkev said:
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!
 
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  • #4
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
 
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  • #5
Math_QED said:
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.
 
  • #6
What an amazing community.
Thanks everyone for the replies! I'll take it from here
 
  • #7
PeroK said:
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.
 

1. What is a subspace in linear algebra?

A subspace in linear algebra is a subset of a vector space that satisfies all the properties of a vector space. In other words, it is a collection of vectors that can be added together and multiplied by a scalar to produce another vector within the same subset.

2. How do you determine if a subset is a subspace?

To determine if a subset is a subspace, you need to check if it satisfies three requirements: closure under addition, closure under scalar multiplication, and contains the zero vector. This means that when you add two vectors from the subset, the result must also be in the subset, and when you multiply a vector from the subset by a scalar, the result must also be in the subset.

3. Why is it important to study subspaces in linear algebra?

Subspaces are important in linear algebra because they allow us to break down complex vector spaces into smaller, more manageable subsets. This makes it easier to study and understand the properties and behaviors of these vector spaces.

4. Can a subspace span the entire vector space?

Yes, a subspace can span the entire vector space. This means that every vector in the vector space can be written as a linear combination of vectors in the subspace. However, this is not always the case. Some subspaces may only span a portion of the vector space.

5. How is a subspace related to the concept of linear independence?

A subspace is related to the concept of linear independence because it is made up of linearly independent vectors. This means that no vector in the subspace can be written as a linear combination of the other vectors in the subset. This is an important property of subspaces as it allows us to perform operations and calculations more easily.

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