Why $L2$ loss is strictly convex if number of samples $N$ is larger than input dimension $d$? I am using $L2$ loss in my linear regression problem and I have to prove that my $L2$ loss is strictly convex if number of samples $N$ is larger than input dimension $d$. 

I think, if I can prove that my hessian matrix, $ X^T X $ for $L2$ loss is positive definite then that will do. Here, $X$ is the feature matrix with  $N$ samples as rows and  $d$ columns.

Also, I understand $v^T X^T X v$ $\ge 0$, 

as,  $v^T X^T X v$ = $(Xv)^T. Xv$ = $|| X.v ||^2_2 $ (Which is the euclidean norm of $X.v$) i.e hessian matrix is positive semi-definite.

But I don't understand how to prove $ X^T X $ is positive definite or $L2$ loss is strictly convex.

-please ignore the constants in $L2$ loss.
I think an additional assumption of linear independent inputs is necessary to have a strictly convex loss function. 
Linear independence with $N&gt;d$ gives you $Xv \neq 0 ~ \forall v \neq 0$, which implies that $X^\top X$ is positive definite, as $||Xv||_2 &gt; 0 $.