Question

Prove that H is a vector space by demonstrating that it satisfies the definition of a vector space or provide an example of a vector in span H that is not in H. (Definition or details of H not provided)

Answer 1

To prove that H is a vector space, we need to demonstrate that it satisfies the definition of a vector space.

Step-by-step explanation:

To prove that H is a vector space, we need to demonstrate that it satisfies the definition of a vector space. A vector space is a set of vectors where addition and scalar multiplication are defined and satisfy certain properties.

1. For any vectors u and v in H, u + v is also in H.
2. For any vector u in H and any scalar c, cu is also in H.
3. There is a zero vector in H that satisfies the property u + 0 = u for every vector u in H.
4. For every vector u in H, there is an additive inverse vector -u such that u + (-u) = 0.
5. The associative property of vector addition holds: (u + v) + w = u + (v + w) for all vectors u, v, and w in H.
6. The commutative property of vector addition holds: u + v = v + u for all vectors u and v in H.
7. The distributive properties of scalar multiplication hold: a(u + v) = au + av and (a + b)u = au + bu, where a and b are scalars and u and v are vectors in H.
8. The associative property of scalar multiplication holds: a(bu) = (ab)u for all scalars a and b and vector u in H.
9. The identity property of scalar multiplication holds: 1u = u for every vector u in H, where 1 is the multiplicative identity.

If we can prove that H satisfies all these properties, then H is a vector space.