Prove, for real $X$, that $\text{rank}(X^TX)=\text{rank}(X)$.
Could anyone please help me with this problem? I've tried to use full-rank factorization and rank-related theorems mentioned in my book but still failed to solve this. I am learning linear algebra by myself and my book has no solutions manual so I find it really hard to get used to solve linear algebra problems.
Many thanks in advance for your help!