Does full rank matrix have a null space?

Question

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 $A$? – twosigma Jul 18 '20 at 19:15 — twosigma, Jul 18 '20 at 19:15

score 11 · Accepted Answer · answered Jul 18 '20 at 19:33

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 $n-m$.

score 5 · Answer 2 · answered Jul 18 '20 at 19:32

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 $m<n$ then the matrix necessarily has larger null space, and if it also has full rank, the null space has dimension $n-m$.

score 2 · Answer 3 · 2020-07-19T03:48:49.187

2

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.

This is true whether or not the matrix has full rank.

edited Jul 19 '20 at 03:48

answered Jul 18 '20 at 23:41

Does full rank matrix have a null space?

3 Answers3