Show that if W and Z are subspaces of V, then (a) their intersection W ∩Z is a subspace of V,(b)

step 1 For A, we have that V intersection W is a subspace of R space to the power of VN, since it conta

Show that if W and Z are subspaces of V, then (a) their...

Show that if $W$ and $Z$ are subspaces of $V$, then (a) their intersection $W \cap Z$ is a subspace of $V,(b)$ their sum $W+Z=\{\mathbf{w}+\mathbf{z} \mid \mathbf{w} \in W, \mathbf{z} \in Z\}$ is also a subspace, but (c) their union $W \cup Z$ is not a subspace of $V$, unless $W \subset Z$ or $Z \subset W$.

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Key Concepts

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same vector addition and scalar multiplication. It must contain the zero vector and be closed under addition and scalar multiplication, which ensures it inherits the structure of the ambient space.

Intersection of Subspaces

The intersection of two subspaces is the set of all elements that are common to both subspaces. Since any element in the intersection satisfies the closure properties of both subspaces, the intersection automatically satisfies the criteria for being a subspace.

Sum of Subspaces

The sum of two subspaces is formed by taking all possible sums of a vector from one subspace and a vector from the other. This construction meets the subspace requirements because the sum of two such combinations is still a combination of vectors from the original subspaces, and scalar multiples preserve the format of these sums.

Union of Subspaces

The union of two subspaces is generally not a subspace since it may not be closed under addition unless one subspace is entirely contained within the other. Without this inclusion condition, the addition of two vectors from different subspaces might produce a vector that does not belong to either subspace.

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