Question

Why does this show that H is a subspace of R³?

A. The vector v spans both H and R³, making H a subspace of R³.
B. The span of any subset of R³ is equal to R³, which makes it a vector space.
C. It shows that H is closed under scalar multiplication, which is all that is required for a subset to be a vector space.
D. For any set of vectors in R³, the span of those vectors is a subspace of R³
E. It shows that H contains the zero vector, which is all that is required for a subset to be a vector space.
F. The set H is the span of only one vector. If H was the span of two vectors, then it would not be a subspace of R³

Answer 1

The correct answer is E. It shows that H contains the zero vector, which is all that is required for a subset to be a vector space.

Step-by-step explanation:

The correct answer is E. It shows that H contains the zero vector, which is all that is required for a subset to be a vector space.

To show that H is a subspace of R³, we need to prove that it satisfies three conditions:

1. It contains the zero vector.
2. It is closed under vector addition.
3. It is closed under scalar multiplication.

Option E states that H contains the zero vector, which satisfies the first condition. The other options do not provide enough evidence to support that H satisfies all three conditions.