I am following Gilbert Strang's Linear Algebra course (link).
In lecture 33, he mentions that a rectangular matrix can’t have a two-sided inverse because either that matrix or its transpose has a nonzero null space. I know the condition for the null space for a square matrix to be invertible (null space must contain only the zero vector). However, I am not able to figure out how the null space comes in when we are talking about left and right inverses.
Referring to the example given in the course lecture: We take an $m\times n$ matrix $A$ with $m>n$. $A$ has full column rank. We can compute the left inverse as $A_{left}^{-1} = (A^TA)^{-1}A^T$ using the invertibility of $A^TA$. He mentions that the right inverse will not exist. I think this has to do something with $N(A^T)$ being nontrivial, but I am not able to figure out the exact relation.
Could someone please explain the relation between invertibility for rectangular matrices and its relation between the two null spaces?
If $=0$ and $=$ then $=()=0=0$ but also $=()==$