functional analysis - 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!!