Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 3

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Corollary to Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations

Let $S$ be a system of homogeneous simultaneous linear equations:

$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$

If $m < n$, then $S$ has at least one non-trivial solution.


Proof

Let $\begin {pmatrix} \mathbf A & \mathbf b \end {pmatrix}$ denote the augmented matrix of $S$.

Because $S$ is homogeneous, we have that $\mathbf b = \mathbf 0$, and so its augmented matrix is $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.


Let $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be a reduced echelon matrix derived from $\begin {pmatrix} \mathbf A & \mathbf 0 \end {pmatrix}$.

Let $S'$ be the system of simultaneous linear equations:

$\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \rho_{i j} x_j = 0$

whose augmented matrix is $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.

By Corollary 1, $S$ and $S'$ are equivalent.

Hence any every solution to $S'$ is also a solution to $S$.


Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.

Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_l$.

Let the remaining columns be named $j_{l + 1}, j_{l + 2}, \ldots, j_n$.

Then we have that $S'$ can be expressed as:

\(\ds x_{j_1} + \sum_{k \mathop = l + 1}^n \rho_{i j_k} x_{j_k}\) \(=\) \(\ds 0\)
\(\ds x_{j_2} + \sum_{k \mathop = l + 1}^n \rho_{2 j_k} x_{j_k}\) \(=\) \(\ds 0\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds x_{j_l} + \sum_{k \mathop = l + 1}^n \rho_{2 j_k} x_{j_k}\) \(=\) \(\ds 0\)

where $l + 1 \le m$.


Setting arbitrary values to $x_{j_k}$ for $l < k \le n$ gives us a non-trivial solution for $S$.

$\blacksquare$


Sources

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