Null Spaces, Column Spaces, and Linear Transformations

What is a null space?

A null space of an mxn matrix A, written as Nul A, is the set of all solutions to the equation Ax = 0. So it's all x such that Ax = 0. 

Theorem: A null space is a subspace

More specifically, the set of all solutions to a system Ax = 0 of m homogeneous linear equations in n unknowns is a subspace.

Fact: Null space is {0} iff the columns of A are linearly independent

What is a column space?

A column space is the Span of a matrix A. 

Theorem: A column space is a subspace

Fact: The column space is all of R^m iff Ax = b has a solution for each b in R^m.