I am sure there is an easy way to reply to this. however I am getting a bit confused by this:
Let v= the vector (a, b) (I can't input the matrix array tonight for some reason). Describe the set H of vectors (x,y) that are orthogonal to v. [Hint: Consider v=0, and v not = 0]
if it is a six marker you may be expected to be very fastidious. Now i am not sure if (0,0) is considered to be orthogonal to a non-zero vector although its dot prod with any other vector will be zero. If not, then from the points of the straight line you have to exclude out the origin. If yes then party anyway.
also see denevo's post.
the cases v = 0, and v ≠ 0 produce two very different kinds of sets for H. this is important.
if v = (a,b) ≠ (0,0), then the space span{v} is a line. think about what it means for a vector to be perpendicular (orthogonal) to a line.
however, if v = (0,0), then span{v} is just the origin. every vector is by definition perpendicular (orthogonal) to the origin. look:
(x,y).(0,0) = x*0 + y *0 = 0 + 0 = 0.