Proof: By Induction

Let $$\alpha_{ij}\in F$$ and $$F$$ be an field, and let $\begin{array}{ccl} \alpha_{11}X_1+\ldots+\alpha_{1n}X_n&=&0\\ \alpha_{21}X_1+\ldots+\alpha_{2n}X_n&=&0\\ \vdots&\vdots&\vdots\\ \alpha_{m1}X_1+\ldots+\alpha_{mn}X_n&=&0\\ \end{array}~~~~~~~~~~~~~~~~ ( * )$ be a homogeneous system of linear equations with $$n$$ unknowns $$X_1,\ldots,X_n$$ and ($$1\le m < n$$) equations. We have to show that the equation system $$( * )$$ has a non-trivial solution.

First, we observe the following:

1. It is sufficient to assume $$m=n-1$$ since if we have found a non-trivial solution for this case, it is also a solution of the more general case $$m < n - 1$$ (after removing some more equations from the system).
2. We can assume $$\alpha_{11}\neq 0$$. If in $$( * )$$ all coefficients $$\alpha_{ij}$$ were zero, then any elements of the field $$F$$ replacing the unknowns $$X_1,\ldots,X_n$$ would be a non-trivial solution. If $$\alpha_{11}= 0$$ but some other $$\alpha_{ij}\neq 0$$, we can change the order of the equations by switching the $$i$$-th equation with the $$1$$-st equation and re-numbering the $$j$$-th unknown with the $$1$$-st unkonwn, so that $$\alpha_{11}\neq 0$$. Note that these operations do not change the non-trivial solution, if it exists.
3. We can assume that besides $$\alpha_{11}$$, all other coefficients in the first column equal zero $$\alpha_{21}=\alpha_{31}=\ldots=\alpha_{m1}=0$$. In order to transform the equation system into this form, we have (for $$i=2,\ldots,m$$) to multiply the first equation with $$\alpha_{i1}$$ and subtract the resulting product form the $$\alpha_{11}$$ multiple of the $$i$$-th equation. Note that these operations do not change the non-trivial solution, if it exists, either.

After these preparations, it is possible to prove the lemma by induction on the number of unknowns.

Base Case $$n=2$$.

The equation system

$\begin{array}{ccl} \alpha_{11}X_1+\alpha_{12}X_2&=&0\\ \end{array}$ has the solution $$X_1:=\beta_1=-\frac{\alpha_{12}}{\alpha_{11}} X_2$$, $$X_2:=\beta_2\in F$$.

Induction step $$n > 2$$.

Let the fundamental lemma be proven for $$m\ge 2$$ unknowns. We have to find a non-trivial solution of the equation system $\begin{array}{cllcl} \alpha_{11}X_1&+&\alpha_{12}X_2+\ldots+\alpha_{1n}X_n&=&0\\ 0&+&\alpha_{22}X_2+\ldots+\alpha_{2n}X_n&=&0\\ \vdots&&\vdots&\vdots&\vdots\\ 0&+&\alpha_{m2}X_2+\ldots+\alpha_{mn}X_n&=&0\\ \end{array}~~~~~~~~~~~~~~~~~~~~~(1)$ According to the base case, there exist a non-trivial solution $$b_2,\ldots,b_n\in F$$ of the equation system $\begin{array}{lcl} \alpha_{22}X_2+\ldots+\alpha_{2n}X_n&=&0\\ \vdots&\vdots&\vdots\\ \alpha_{m2}X_2+\ldots+\alpha_{mn}X_n&=&0\\ \end{array}~~~~~~~~~~~~~~~~~~~~~(2)$ Note that we have renumbered the unknowns of the smaller equation system $$(2)$$ from $$X_1,\ldots,X_{n-1}\in F$$ to $$X_2,\ldots,X_n\in F$$ in order to better embed them into the greater equation system $$(1)$$. Then the solution of the greater equation system is $$X_1:=\beta_1=-\frac{\alpha_{12}}{\alpha_{11}} X_2 -\ldots -\frac{\alpha_{1n}}{\alpha_{11}} X_n$$, $$X_i:=\beta_i\in F$$, $$i=2,\ldots,n$$.

Thank you to the contributors under CC BY-SA 4.0!

Github:

References

Bibliography

1. Koecher Max: "Lineare Algebra und analytische Geometrie", Springer-Verlag Berlin Heidelberg New York, 1992, 3rd Volume