Answer :
The intersection of two subspaces V and W is indeed a subspace of Rⁿ, as it satisfies all subspace properties like non-emptiness, closure under addition, and scalar multiplication.
Nevertheless, the union of two subspaces is not always a subspace, as showed with an example consisting of two dimensional subspaces in R².
Subspaces, intersection, and union are foundational concepts in sets and vector spaces. Let's look at how they function within R^n.
a) To prove that the intersection of V and W (denoted as V∩W) is a subspace of R^n, we need to show it satisfies three properties of subspaces:
- It must be non-empty (0 is in V∩W).
- All sums of vectors in V∩W belong to V∩W.
- All scalar multiples of vectors in V∩W belong to V∩W.
These all hold because by definition, intersection includes elements found in both V and W. Thus, if these elements belong to both subspaces, which satisfy subspace properties, the overlap will too.
b) However, union of two subspaces (V∪W) isn't necessarily a subspace. Consider two one-dimensional subspaces in R^2: V along the x-axis, and W along the y-axis. Their union doesn't contain the zero vector (unless one or both subspaces are the zero subspace), and hence doesn't satisfy necesarry subspace properties. So, union of two subspaces is not always a subspace.
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