# Thread: A transposed problem A^T

1. ## A transposed problem A^T

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

2. Originally Posted by bugal402
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

What is N(A), C(A)...??

Tonio

3. n(a) is null space

c(a) is column space

4. Originally Posted by bugal402
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)
If $Ax=0$ then obviously $A^{\textsc t}Ax=0$. Therefore $\mathbf{N}(A)\subseteq\mathbf{N}(A^{\textsc t}A)$.

For the converse, suppose that $A^{\textsc t}Ax=0$. Then $(Ax)^{\textsc t}(Ax) = x^{\textsc t}A^{\textsc t}Ax =0$, which implies that $Ax=0$.

For B), use the rank+nullity theorem.