Final answer:
To determine which subsets of R³ × ³ are subspaces of R³ × ³, we need to check if they satisfy the three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector.
Step-by-step explanation:
In order to determine which subsets of R³ × ³ are subspaces of R³ × ³, we need to check if they satisfy the three properties of a subspace: closure under addition, closure under scalar multiplication, and containing the zero vector.
For example, the subset consisting of the zero vector alone is a subspace of R³ × ³ because it satisfies all three properties. Another example is the subset of all vectors where the third component is equal to zero, which is also a subspace. However, a subset where the sum of the first and second components is equal to 1 would not be a subspace because it does not satisfy the closure under addition property.
Therefore, the subsets that are subspaces of R³ × ³ are the subset consisting of the zero vector alone and the subset where the third component is equal to zero.