The first question revloves around this one from the text:

Let $T: \mathbb{R}^n \to \mathbb{R}$ be a linear map. Prove that there is a vector $\mathbf{a} \in \mathbb{R}^n$ so that $T(\mathbf{x}) = \mathbf{a} \cdot \mathbf{x} $

Usually for an existence question I have to show the existence of the object. In this case that object is $\mathbf{a}$, which means I have to construct it. Or is it enough to show that a linear map in this case defined as $T(\mathbf{x}) = \mathbf{a} \cdot \mathbf{x}$ satisfies the "usual conditions of a linear map? i.e:

If $A$ is an $m \times n$ matrix. Then $\|A\| \leq \sqrt{\sum_{i,j}a_{i,j}^2} \leq \sqrt{n} \|A\|$

so $Ax$ is going to be an $m \times 1$ vector of which we take the norm in this case it will be the unit vector that gives maximum norm. If we said $m = 2$ it will be $\sqrt{a_1^2 + a_2^2}$. Now assuming I understood the notation correctly $\sqrt{\sum_{i,j}a_{i,j}^2} = \sqrt{a_{1,1}^2 + a_{1,2}^2 + a_{2,1}^2 + a_{2,2}^2 }$. That part of the inequality makes sense.

But the second bit is throwing me off because if I take the same $\|A\|$ I still have the same $m \times 1$ vector I'm taking the…