A subspace is a subset of a vector space that contains all of the properties of the original vector space. This means that it must be closed under addition and scalar multiplication and must contain the zero vector.
To determine if H is a subspace of V in P3, you must check if it meets the three criteria of a subspace. These criteria are: closure under addition, closure under scalar multiplication, and containing the zero vector.
Closure under addition means that if you take any two vectors from the subspace and add them together, the resulting vector must also be in the subspace. In other words, the sum of any two vectors in the subspace must also be in the subspace.
Closure under scalar multiplication means that if you take any vector from the subspace and multiply it by any scalar, the resulting vector must also be in the subspace. In other words, the scalar multiple of any vector in the subspace must also be in the subspace.
Determining if H is a subspace of V in P3 is important because it ensures that all of the properties and operations of a vector space can be applied to the subspace. This is necessary for performing calculations and solving problems involving vectors in the subspace.