Suppose that A is a 3 x 3 matrix whose nullspace is a line through the origin in 3-space. Can the row or column space of A also be a line through the origin? Well if we have a matrix whose nullspace is a line through the origin then we have.. [X;Y;Z] = [a;b;c] t And we know that the dimensions of our nullspace is 1.. Also we know that nullspace(a)+rank(a) = n where n is the columns. so.. 1+rank(a)=3 this implies the rank = 2. Which has a dimension of two, which means it is a plane through the origin. If the dimension was one, then we would have a line through the origin.