H U K is a subspace iff H is contained in K or K is contained in H Proof

TL;DR

The union of subspaces in a vector space is a subspace if and only if one subspace is contained in the other.

Transcript

okay we have to show that h union k is a subspace of a vector space v if and only if h is contained in k or k is contained in h and all of this will assume that k and h themselves are also subspaces of a vector space v but i just i forgot to write it so proof all right let's prove this direction first so we'll start by assuming so suppose that h is... Read More

Key Insights

• 👾 Subspaces in a vector space play a crucial role in determining the properties of their unions.
• 🤩 Containment of one subspace within another is a key factor in determining if the union forms a subspace.
• 👍 The assumption of subspace properties is essential in proving or disproving the subspace nature of the union.

Questions & Answers

Q: How does the proof establish that the union of h and k is a subspace?

The proof shows that if h is contained in k or k is contained in h, the union of h and k is itself a subspace. By examining these containment scenarios, the proof demonstrates the subspace property of the union.

Q: What contradiction is reached when assuming h union k is a subspace without one subspace being contained in the other?

By assuming that h union k is a subspace without h being contained in k or vice versa, the proof shows that there exists a contradiction in the sum of elements from h and k not being in the union, thus proving the union is not a subspace.

Q: How does the proof utilize concepts of vector spaces to establish the properties of subspace unions?

The proof leverages the definitions and properties of vector spaces, including containment and subspaces, to logically demonstrate the conditions under which the union of subspaces forms a subspace itself.

Q: Why is proving the contrapositive an effective strategy in showing that h union k is not a subspace under certain conditions?

Proving the contrapositive allows for a direct approach to demonstrate that if h union k is not a subspace due to the lack of containment between h and k, a contradiction arises when examining the sum of elements from both subspaces.

Summary & Key Takeaways

• The proof shows that the union of subspaces in a vector space is itself a subspace if one subspace is contained in the other.

• If both subspaces are the same, the union is a subspace.

• The proof involves showing that the sum of elements from each subspace is not in the union, leading to a contradiction.