In a book I am reading orthogonality of the null space and the row space is proven as follows:
For any $x\in ker(A)$ and any y satisfying $A^Ty$ $$x^T(A^Ty)=(Ax)^Ty=0^Ty=0$$ The step that I am not clear on is why $x^T(A^Ty)=(Ax)^Ty$? First why is the transpose distributive and second why does A appear on the right on the right hand side of the equation and on the left on the left hand side?
@SahibaArora has linked you to a proof on this site that $x^TA^T=(Ax)^T$. By associativity, $x^T(A^Ty)=(x^TA^T)y=(Ax)^Ty$.