A subspace is a subset of a vector space that satisfies the same properties as the original vector space. This means that it is closed under addition and scalar multiplication.
To show that a set is a subspace, you need to prove that it is closed under addition and scalar multiplication. This can be done by showing that the set contains the zero vector, is closed under addition, and is closed under scalar multiplication.
A subset is a collection of elements that are contained within a larger set. A subspace, on the other hand, is a subset of a vector space that satisfies the same properties as the original vector space.
Yes, a set can be a subspace of multiple vector spaces as long as it satisfies the properties of a subspace for each vector space it is a part of.
Proving that a set is a subspace is important because it allows us to understand the properties and behavior of that set within a larger vector space. It also helps us to perform calculations and make predictions about the set with confidence.