Let a be a fixed nonzero vector in R-n.

(a) Show that the set of S of all vectors x such that a dot x = 0 is a subspace of R-n.

(b) Show that if k is a nonzero real number, then the set A of all vectors x such that a dot x = k is not a subspace.

Thanks in advance.

Question

Let a be a fixed nonzero vector in R-n.

(a) Show that the set of S of all vectors x such that a dot x = 0 is a subspace of R-n.

(b) Show that if k is a nonzero real number, then the set A of all vectors x such that a dot x = k is not a subspace.

Thanks in advance.

anonymous · Answer

Well (a) is just the nullspace of the vector space characterized by A. 
(b) is solving a system of equations in which the space characterized is not closed under addition. I think.