Final answer:
H, a subset of the 3-dimensional vector space V defined by the equation 3x+7y-7z=21, is a subspace because it is nonempty, contains the zero vector, and is closed under vector addition and scalar multiplication.
Step-by-step explanation:
The subject of the student's question deals with the subset H of a vector space V in three dimensions and whether H can be considered a subspace. To determine if H is a subspace, we check if it is nonempty, contains the zero vector, and is closed under vector addition and scalar multiplication. Since the equation of the plane 3x+7y−7z=21 can be written as z = (3x+7y-21)/7, we can see that H is nonempty, and the zero vector (0,0,0) is a solution (when x=y=0, z=0), hence, H contains the zero vector.
The plane's equation can also be satisfied by the linear combinations of any two vectors in the plane, therefore, the points on the plane H satisfy the conditions for vector addition and scalar multiplication needed for a subset to be a subspace. Thus, H qualifies as a subspace of V.