Prove $X^tX$, where $X$ is a matrix of full column rank, is positive definite?

Question

Let $X$ be a matrix of dimension $n\times k$ where $n>k$, $\text{rk}(X)=k$ so $X$ is of full column rank. Then how do I prove $X^tX$ is always positive definite, where $X^t$ is transpose of $X$? This is given sortta like a lemma in our lecture slides without proof but would like to have some reasoning behind this. Thank you for your help!

Answer 1

For any real invertible matrix $X$ we can show that the product $X^tX$ is a positive defined matrix. In fact, let's just take a vector $v$ non-zero. So we can easily see that: $$v^tX^tXv=||Xv||^2>0$$ because since the matrix $X$ is invertible, $Xv\neq 0.$