Final answer:
The only value for α such that {0} is a subspace of the real numbers, fulfilling the subspace requirements of including the zero element, and being closed under addition and scalar multiplication.
Step-by-step explanation:
The student is asking for the value of α such that {α} forms a subspace of ℝ, the set of all real numbers. In mathematics, a subspace is a subset of a vector space that is itself a vector space under the same operations of vector addition and scalar multiplication.
For a set to be a subspace of ℝ, it must include the zero vector (which is 0 in the context of real numbers), be closed under vector addition and scalar multiplication. Considering these conditions, the only subset of real numbers that forms a subspace is the set containing only the zero element, often denoted as {0}. Therefore, α must be 0.
In order for (α) to be a subspace of ℝ, it must satisfy three conditions: closure under addition, closure under scalar multiplication, and contain the zero vector.
To find α, we can choose any set of vectors that meet these conditions. For example, we can choose α = {0}, which contains only the zero vector. Since the zero vector is always contained in any vector space, α = {0} is indeed a subspace of ℝ.