Subspace Verification of Vectors of Form x 0 x

Let V be a vector space. Verify whether a subset S, which is made of vectors of the form (x, 0, -x), is a subspace of V by checking the properties of closure.

When determining if a subset of a vector space is a subspace, it is crucial to check for two fundamental properties: closure under addition and closure under scalar multiplication. These properties ensure that any linear combination of the vectors within the subset remains within the subset, maintaining the structural integrity of a vector space. For the subset given in this problem, which includes vectors of the form (x, 0, -x), consider how any two such vectors are added and how they are affected by scalar multiplication. This involves proving that for any vectors u and v in the subset, as well as any scalar c, the vector cu is also in the subset and so is u + v. This analysis reinforces the importance of understanding subspace properties and highlights the elegance of linear algebra in defining and manipulating vector spaces. Essentially, you are establishing an internal consistency that aligns with the vector space axioms, ensuring that the subset behaves like a vector space within the larger vector space V.