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A **subspace** of a vector space is a subset that behaves like a smaller vector space within it. To qualify as a subspace, a set must meet specific criteria:

• It includes the zero vector of the larger vector space.
• It is closed under vector addition, meaning adding any two vectors in the subspace results in another vector inside the same subspace.
• It is closed under scalar multiplication, meaning scaling any vector in the subspace by a number (scalar) keeps it within the same subspace.

These criteria ensure that the operations and structure of the vector space remain consistent within the subspace.

The **intersection of subspaces** is the set of vectors that belong to both subspaces. For example, if we have two subspaces, say \( H \) and \( K \), of a vector space \( V \), then their intersection \( H \cap K \) consists of all vectors common to both \( H \) and \( K \).To check that this intersection \( H \cap K \) forms a subspace, we verify the three subspace criteria:

• The zero vector is present in both subspaces, thus, it is in the intersection.
• Adding any two vectors from the intersection keeps the result within both subspaces, due to closure properties of these subspaces.
• Scaling any vector in the intersection is valid and keeps it within both \( H \) and \( K \), adhering again to the scalar multiplication property.

Therefore, \( H \cap K \) fulfills all requirements of being a subspace.

The **union of subspaces** refers to the set of vectors that are in either one of the subspaces, or both. Given subspaces \( H \) and \( K \) in \( V \), their union is \( H \cup K \), containing vectors from either \( H \) or \( K \). Despite containing more vectors than the intersection, this union does not generally form a subspace. Consider the example of two subspaces in \( \mathbb{R}^2 \), the x-axis \( H \) and y-axis \( K \). Their union would include all points on either axis, but not combinations like \((1, 1)\), which is a sum of \((1, 0)\) and \((0, 1)\).Since the union fails to stay within itself under addition or scalar multiplication, it cannot satisfy the subspace criteria, and so it is not a subspace.

**Closure under addition** is a fundamental property of subspaces, ensuring that any addition of two vectors from the subspace results in a vector that still belongs within that subspace. This is crucial, because it preserves the structure of the subspace for addition operations.When verifying closure under addition:

• Take any two vectors from the subspace. For example, in \( H \cap K \), vectors \( \mathbf{u} \) and \( \mathbf{v} \) from \( H \cap K \) will satisfy \( \mathbf{u} + \mathbf{v} \) belonging to both \( H \) and \( K \).

This is true because \( H \) and \( K \) themselves are closed under addition. Consequently, ensuring closure under addition helps verify whether a given subset meets the requirements of being a subspace of a vector space.