## If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible

## Problem 38

Let $A$ be an $m \times n$ real matrix.

Then the* kernel* of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.

The kernel is also called the* null space* of $A$.

Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible.

(*Stanford University Linear Algebra Exam*)