Elementary Row Operation on Augmented Matrix leads to Equivalent System of Simultaneous Linear Equations/Corollary 4
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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:
- $\ds \forall i \in \set {1, 2, \ldots, m}: \sum_{j \mathop = 1}^n \alpha_{i j} x_j = 0$
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 the number of non-zero rows of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$ be $l$.
If $l = n$, then the only solution to $S$ is the trivial solution.
Proof
Consider the structure of $\begin {pmatrix} \mathbf R & \mathbf 0 \end {pmatrix}$.
Suppose the leading coefficients appear in columns which we name $j_1, j_2, \ldots, j_n$.
As there are $n$ columns as well as $n$ non-zero rows:
Thus $S'$ can be expressed as:
\(\ds x_{j_1}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds x_{j_2}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||
\(\ds x_{j_n}\) | \(=\) | \(\ds 0\) |
and the result follows.
$\blacksquare$
Sources
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Corollary $3$