Determine $\| A\|$ when we consider the norms $\|\cdot\|_{1}=\|\cdot\|_\infty$. 
Let $M$ a matrix of $m\times n$. Consider $A$ as a linear map of $\mathbb{K}^n\to\mathbb{K}^m$. Determine $\|A\|$ when we consider the norms $\|\cdot\|_1$ and $\|\cdot\|_\infty$ in $\mathbb{K}^n$ and $\mathbb{K}^m$, respect.

I have the following questions: 
First of all, the linear map $A:\mathbb{K}^n\to\mathbb{K}^m$ has a explicit form?. 
Second, how I understand the $1$-norm  and the infinite norm for a matrix of side $m\times n$??
Can help me in this problem, I try use cauchy inq. thanks!!
 A: Hint 1: As mentioned in the comments, an $m \times n$ matrix $A$ can be viewed as a map that takes a vector $x \in K^n$ and maps it to $Ax \in K^m$.
Hint 2: I think you mean "consider the norms $\|\cdot \|_1$ and $\|\cdot\|_\infty$ in $K^n$ and $K^m$ respect." (the norms are not equal). The definition of $\|A\|$ can be written in various ways.
$$\sup_{x \in K^n : \|x\|=1} \|A x\|_\infty = \sup_{x \in K^n:x \ne 0} \frac{\|A x\|_\infty}{\|x\|_1}.$$
For example see the Wikipedia page (scroll down to the definition of $\|A\|_{\alpha,\beta}$).
In plain words, if you consider all vectors in $K^n$ with $\|\cdot\|_1$-norm equal to $1$, map them using $A$, then take the $\|\cdot\|_\infty$-norm of the result, the largest possible value is what we call the norm of $\|A\|$.
Start with this definition and see if you can finish the problem on your own.

Proof sketch:
\begin{align}
\sup_{x \in K^n : \|x\|=1} \|A x\|_\infty
&= \sup_{x \in K^n : \|x\|=1} \max_i A_{i,\cdot}^\top x
\\
&= \max_i \sup_{x \in K^n : \|x\|=1} A_{i,\cdot}^\top x
\\
&= \max_i \max_j |A_{i,j}|.
\end{align}
Equality is attained when $x = e_j$ where $j$ is the column containing the largest entry of $A$ (in absolute value).
