In Linear Algebra Done Right, it defines the linear maps of a homogeneous system of linear equations with n variables and m equations as
$$T(x_1,...,x_n) = (\sum_{k=1}^{n} A_{1,k}x_k,...,\sum_{k=1}^{n} A_{m,k}x_k) = 0$$
My question is why the homogeneous system of linear equations can be expressed as the linear maps from $\mathbb{F}^n \mapsto \mathbb{F}^m$.
In the book it uses this form to prove "A homogeneous system of linear equations with more variables than equations has nonzero solutions".
A homogeneous system of linear equations is, by definition, of the form $$\begin{align} A_{11}x_1+A_{12}x_2+\dots+A_{1n}x_n &=0\\ &\vdots \\ A_{m1}x_1+A_{m2}x_2+\dots+A_{mn}x_n &=0 \end{align}$$ where all $A_{ij}$ are fixed scalars, i.e. are $\in\Bbb F$ (which is typically $\Bbb R$ or $\Bbb Q$ for the first round).
Now, consider the linear map $T:\Bbb F^n\mapsto\Bbb F^m$ defined by the given formula, i.e. a tuple $(x_1,\dots,x_n)$ of scalars is mapped to the tuple (column vector) given by the left hand sides of the equations, that is, $$T\ :=\ (x_1,\dots,x_n)\mapsto(\sum_kA_{1k}x_k,\dots,\sum_kA_{mk}x_k)$$ You can readily verify that this indeed defines a linear map, and that the set of equations is exactly its kernel $\ker T=\{x\in\Bbb F^n:T(x)=0\}$.
Moreover, and most importantly, if we put the coefficients in a matrix $A$, and regard tuples as column vectors, then we simply have $$\forall x\in\Bbb F^n:\ T(x)=A\cdot x$$