If we have a system of $m$ equations and we have $n $ unknowns where $n>m $, say
$$ \sum_{i=1}^{n} a_{i,j} x_i=0 $$ for $j=1,…,m $, and these are equations in some field $K$.
Of course there is the trivial solution. But how can we say that there is a nontrivial solution?
This is an assumed result in a proof in Galois theory, and I know this is an elementary result, but I can’t find a proof anywhere. Can anyone provide or link a proof of this.
Best Answer
To simplify, suppose that the system is$$\left\{\begin{array}{l}a_{11}x+a_{12}y+a_{13}z=0\\a_{21}x+a_{22}y+a_{23}z=0.\end{array}\right.\tag1$$Then either the matrix $A=\left[\begin{smallmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{smallmatrix}\right]$ is invertible or it isn't. If it is, you take any number $z$, and you take $x,y\in K$ such that$$(x,y)=A^{-1}.(-a_{13}z,-a_{23}z).$$Then $A.(x,y)=(-a_{13}z,-a_{23}z)$; in other words, $(1)$ holds. Of course, if $a_{13}=a_{23}=0$, this will provide you only one solution but, in that case, $(0,0,z)$ is a solution for every $z\in K$.
And if $A$ is not invertible, let $v=(\alpha,\beta)\in\ker A\setminus\{(0,0)\}$. Then, for each $\lambda\in K$, $(\lambda\alpha,\lambda\beta,0)$ is a solution of $(1)$.
If you are still looking for a reference, then that's the LEMMA ON LINEAR EQUATIONS from Nathan Jacobson's Basic Algebra I, 2nd edition, chap. 4, § 5, p. 237.