Question

Let A be a matrix. show that if x within N(A^T A), then Ax is in both the Column Space of A { Col (A)} and the Null Space of A transpose {N(A^T)}.

Now show N(A^T A) = N(A). Conclude that if A has linearly independent columns, then A^T A is invertible.

Answer 1

Ax is just a linear combination of the columns of A so its in the column space of A. since x is a null space of A^TA then we know,
A^TAx=0
So,
A^T(Ax)=0
now we see that Ax is in the null space of A^T since A^T(Ax)=0

the N(A^T A) is the set of all vectors x such that A^T Ax=0. Notice that A^T will only be zero if Ax=0.... So we can see that A^T will only be zero if Ax=0, which will only happen if x will be in the null space of A. Now we can conclude that set of all vectors x which will make Ax=0 will also make A^TAx=0, and so N(A^TA)=N(A).

A^T A will be invertible if A^T Ax=0 if it has only the trivial solution(i.e. x=0) now since A has linearly independent columns then the only solution for Ax=0 is the trivial solution x=0. So A^TAx=0 will have only the trivial solution x=0, therefore, it will be invertible