# Does full rank matrix have a null space? [closed]

The null space is defined as all vector that is set to null by matrix $$A$$, where $$Ax = 0$$.

If the matrix $$A$$ is full rank, does it mean that it has no null space?

• What is $Ax$ for $x = 0$ and any matrix $A$? Commented Jul 18, 2020 at 19:15

An $$m\times n$$ matrix has full rank if it has the maximum rank possible. When $$m=n$$, of course, this means the matrix is invertible. When $$m>n$$, this means it has rank $$n$$ and the nullspace consists just of $$0$$. However, when $$n>m$$, this means that the matrix has rank $$m$$ and the nullspace will have dimension $$n-m$$.
Any matrix always has a null space. An $$m\times n$$ full rank matrix with $$m\geq n$$ has only the trivial null space $$\{0\}$$. If $$m then the matrix necessarily has larger null space, and if it also has full rank, the null space has dimension $$n-m$$.
It's straight forward to check that the "null space", $$\{x:Ax=0\}$$, is always a vector space. As such it always contains the zero vector. Indeed, any linear transformation takes zero to zero.