Can we conclude that this matrix is definite positive? Let $A$ be a $n\text{-by-}m$ matrix. Suppose that columns of $A$ are linearly independent. Can we conclude that $A^TA$ is definite positive? Could you help me with proof? Thanks.

Observe that for any $\xi \in \mathbb{R}^m$, $$ \langle \xi, A^T A \xi \rangle_{\mathbb{R}^m} = \langle A \xi, A \xi \rangle_{\mathbb{R}^n} = \|A \xi \|^2_{\mathbb{R}^n}. $$ Now:

1. What do you know about the sign of $\|A \xi \|^2_{\mathbb{R}^n}$?
2. Given that the columns of $A$ are linearly independent, when is it possible for $\|A \xi \|^2_{\mathbb{R}^n} = 0$?