Consider the ordinary vectors in three dimensions (axi^+ayj^+azk^), with complex components.
(a) Does the subset of all vectors with az=0 constitute a vector space? If so, what is its dimension; if not, why not?
(b) What about the subset of all vectors whose z component is 1? Hint: Would the sum of two such vectors be in the subset? How about the null vector?
(c) What about the subset of vectors whose components are all equal?