System of homogeneous linear equations with coefficients in a field In the book I started to read, at almost the very beginning it is stated (without proof):

If $m<n$ and we have a system $$a_{11}x_1+...+a_{1n}x_n=0$$
$$\vdots$$
$$a_{m1}x_1+...+a_{mn}x_n=0$$
of homogeneous linear equations, with coefficients in a field, then there is a non-trivial solution in the field.

So, I have some questions about this:
Where a proof can be found? Does this also holds for finite fields (or it is implicitly assumed that we talk about infinite fields)? Is this just an application of matrix algebra (that is, this matrix of coefficients will always have an inverse if $m<n$?

This is true over any field, and is just basic linear algebra.  Let $F$ be the field in question, and let $T:F^n\to F^m$ be the linear map given by the matrix $(a_{ij})$.  Since $n>m$, the dimension of the domain of $T$ is greater than the dimension of the codomain, so $T$ cannot be injective (if you like, by the rank-nullity theorem, the nullity of $T$ is at least $n-m$ since the rank of $T$ is at most $m$).  So the kernel of $T$ is nontrivial, which means exactly that there is some $(x_1,\dots,x_n)$, not all $0$, which is a solution to your system of equations.
