Final answer:
The set S defined by 5x1+6x2=0 in the vector space V = R₂ is a subspace, as it contains the zero vector, is closed under vector addition, and is closed under scalar multiplication, thus meeting all the necessary conditions of a subspace.
Step-by-step explanation:
The student asked to determine whether a given set S is a subspace of the vector space V, where V = R₂, and S is the set of all vectors (x₁,x₂) in V satisfying 5x₁+6x₂=0.
To be a subspace, a set must satisfy three conditions: it must contain the zero vector, it must be closed under vector addition, and it must be closed under scalar multiplication. Let's check each condition with respect to set S:
Since set S satisfies all three subspace conditions, we can conclude that S is indeed a subspace of vector space V.