Suppose A is an n × K matrix with full column rank of K. Show that
A'A (K×K) is nonsigular (invertible), which means Rank(A'A)=Rank(A)=K.

Prove equality of their null spaces. Null space of the A'A matrix is given by vectors x for which A'Ax=0. If this condition is fulfilled, also holds 0=x'A'Ax=0, which means Ax=0.

On the other hand, if Ax=0, then A'Ax=0.

So, their null spaces are same, and therefore Rank(A'A)=Rank(A)=K.