Proof: U ∪ W is a Subspace of V if and only if U ⊆ W or W ⊆ U | Linear Algebra Explained

In this video, we explore a fascinating result in linear algebra: Let U and W be subspaces of a vector space V. Prove that the union of U and W (U ∪ W) is a subspace of V if and only if U ⊆ W or W ⊆ U. This theorem highlights the conditions under which the union of two subspaces forms a valid subspace.

📌 Topics Covered:
Review of subspaces and their properties
Explanation of the union of two sets in the context of vector spaces
Step-by-step proof of the "if and only if" condition:
Proving that if U ∪ W is a subspace, then U ⊆ W or W ⊆ U
Proving that if U ⊆ W or W ⊆ U, then U ∪ W is a subspace
Examples to illustrate why this condition is necessary
Discussion on the relationship between unions, intersections, and subspaces
By the end of this video, you’ll have a deep understanding of why the union of two subspaces is generally not a subspace and what makes it a subspace in specific cases.

👉 Perfect for students tackling subspaces in linear algebra or anyone curious about vector space theory!

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By Mexams