Theorem 9.28 Rudin

Question

$A(h,k)$ = $A_xh$ + $A_yk$.

Where $A_xh$ $\in$ $L(R^n)$, $A_yk$ $\in$ $L(R^m,R^n)$.

I don't understand that how does it comes from $f(a,b)$ = 0, that $f(a+h,b+k)$ = $A(h,k) + r(h,k)$ Hence I Couldn't understand the last equation on the photo .

Any help would be appreciated.

Answer 1

That is simply because $$ f(a+h,b+k)=f(a,b)+A(h,k)+r(h,k) $$ which follows from the definition of $A$ and $f'(a,b)$.