Is this a basis for the Bergman space?

What is the difference between the basis of a null space and the dimension of a null space for a matrix?

• I'm rather confused and I'm starting to think it's just the same thing described with different words. If I get the *dimension of the null space*, I'm looking for a number such as 2, right? If I get the *null space* of a matrix, I get the set of vectors that when added together give me the zero matrix, right? What does the basis of a null space mean then? And what does the basis of a column space mean?

• Answer:

  It seems that your confusion is coming from the term "basis." A basis (in this context) is a linearly independent set of vectors which spans a space. Consider R^2. A basis for R^2 is {(0,1), (1,0)}. These vectors are linearly independent since if x(0,1) + y(1,0) = (0,0), that implies that (0,x) + (y,0) = (0,0) which implies that (y,x) = (0,0) and so x = y = 0. Therefore, this set is linearly independent. The set spans the space R^2 since for any vector (x, y), we can write it as a linear combination of the vectors (in this case... x(1,0) + y(0,1)). So the basis of a null space is a linearly independent set of vectors which spans the null space. And yes, the dimension of a space is a number, such as 2. The dimension is equal to the number of elements in a basis (all bases of a space are guaranteed to have the same amount of vectors).