let A be an m x n matrix...prove that N(A^T*A)=N(A) problem gives a hint vector x is a member of N(A^T*A) then A*vector x is a member of C(A) n N(A^T)
B) prove that rank (A)=rank(A^T*A)
C) prove that C(A^T*A)=C(A^T)
Thanks
If $\displaystyle Ax=0$ then obviously $\displaystyle A^{\textsc t}Ax=0$. Therefore $\displaystyle \mathbf{N}(A)\subseteq\mathbf{N}(A^{\textsc t}A)$.
For the converse, suppose that $\displaystyle A^{\textsc t}Ax=0$. Then $\displaystyle (Ax)^{\textsc t}(Ax) = x^{\textsc t}A^{\textsc t}Ax =0$, which implies that $\displaystyle Ax=0$.
For B), use the rank+nullity theorem.