# 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.