# Prove that to every $A \in L(R^n,R^1)$ corresponds a unique $y \in R^n$ such that Ax=xy. Prove that $||A||=|y|$

Prove that to every $$A \in L(R^n,R^1)$$ corresponds a unique $$y \in R^n$$ such that $$Ax=\langle x, y\rangle$$. Prove that $$\|A\|=|y|$$

Definition: For $$A \in L(R^n,R^m)$$, define the norm $$\|A\|$$ of A to be the sup of all numbers $$|Ax|$$, where x ranges over all vectors in $$R^n$$ with $$|x| \leq 1$$. The inequality $$|Ax| \leq \|A\| \cdot |x|$$ holds for all $$x \in R^n$$.

I am thinking about induction on the power of $$n$$, but not sure how to express each individual $$A$$, $$x$$ and $$y$$.

For a proof avoiding bases, define a map $$T:R^n\to L(R^n, R)$$ by sending $$y\in R^n$$ to $$\langle \cdot, y\rangle$$. If $$y\in \ker(T)$$ then, in particular, $$\langle y, y\rangle =0$$, and hence $$y=0$$. So $$\ker T= 0$$. By the rank-nullity theorem, we see that $$T$$ is a linear isomorphism. Thus for every $$A\in L(R^n, R)$$ there is $$y\in R^n$$ with $$Ax=\langle x, y\rangle$$ for all $$x$$.
(Of course, the basis-free proof uses bases in disguise in the rank-nullity theorem and in the fact that $$\dim L(R^n, R) = n$$).
Hint: What does $$A$$ do to each standard basis vector $$e_k$$?
Hint: Consider a base of $$R^n$$ like $$\{e_i\}$$ and apply $$A$$ to each $$e_i$$, assuming such $$y$$ exists. What should be this $$Ae_i$$?