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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?

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    $\begingroup$ What is $Ax$ for $x = 0$ and any matrix $A$? $\endgroup$
    – twosigma
    Jul 18, 2020 at 19:15

3 Answers 3

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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 $n-m$.

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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 $n-m$.

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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.

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