Why is $x^T(A^Ty)=(Ax)^Ty$?

Question

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?

Generally for matrices $(AB)^T=B^TA^T$ – J. W. Tanner Jul 03 '20 at 20:58 — J. W. Tanner, Jul 03 '20 at 20:58

score 1 · Answer 1 · edited Jul 03 '20 at 21:10

1

@SahibaArora has linked you to a proof on this site that $x^TA^T=(Ax)^T$. By associativity, $x^T(A^Ty)=(x^TA^T)y=(Ax)^Ty$.

edited Jul 03 '20 at 21:10

J. W. Tanner

60,406

answered Jul 03 '20 at 20:56

J.G.

115,835

Why is $x^T(A^Ty)=(Ax)^Ty$?

1 Answers1