Linear Algebra/Homogeneous Systems

A homogeneous system of linear equations are linear equations of the form

$$a_{11}x_1+a_{12}x_2+a_{13}x_3+\ldots+a_{1n}x_n=0$$

$$a_{21}x_1+a_{22}x_2+a_{23}x_3+\ldots+a_{2n}x_n=0$$

$$a_{31}x_1+a_{32}x_2+a_{33}x_3+\ldots+a_{3n}x_n=0$$

$$\vdots$$

$$a_{m1}x_1+a_{m2}x_2+a_{m3}x_3+\ldots+a_{mn}x_n=0$$

The trivial solution is when all xn are equal to 0.

Consider the matrix of coefficients

$$A=\left(\begin{matrix} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn} \end{matrix}\right)$$

A linear combination of the columns of A where the sum is equal to the column of 0's is a solution to this homogeneous system. A solution where not all xn are equal to 0 happens when the columns are linearly dependent, which happens when the rank of A is less than the number of columns. However, if the rank is equal to the number of columns, then all the columns are linearly independent, so then the only solution is the trivial one.