Question

What is the difference between a basis for a null space (A) and the span (A)?

Answer 1

they are entirely different.
an excellent example is here
http://www.youtube.com/watch?v=_uTAdf_AsfQ

Answer 2

I think I am even more confused than what I originally was. The question in my book asks me to find a basis for the null space of this given matrix.

Now, I understand how to find the null space. Row reduced and rewrite as a combination.


Now since the null space in not only the zero vector. The original column space of A is dependent. And both x3 and x4 can be removed to find a new column space of A with only the first two vectors. But To be a basis it has to be independent, and if you row reduce A to find a null space and it was dependent, then wouldn't the resulting null space always be a basis? and it is always a basis how is that any different than asking to find the span of A to begin with?