I am stuck with the following question.:
Suppose $A \in \mathbb{R}^ {m \times n} $ has a zero null space. Show that $A$ is left invertible and $A^TA$ is positive definite.
How do I show that $A$ has a left inverse without using the linear transformation perspective i.e., $A \in \mathbb{R}^{m \times n} \Rightarrow \exists T: \mathbb{R}^n \to \mathbb{R}^m$ given by $x \mapsto Ax$.
I just want to use basic matrix theory stuffs like SVD etc. prove the existence of left inverse of $A$