Section 2 Theorems

Theorem 1 - Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to exactly one reduced echelon matrix.

Theorem 2 - Existence and Uniqueness Theorem

A linear system is consistent if and only if the rightmost column of the augmented matrix is pivot column - that is, if and only if an echelon form of the augmented matrix has no row of the form

\begin{align} [0 \text{ } ... 0 \text{ } |b ] \end{align}

with b nonzero

If a linear system is consistent, then the solution set contains either (i) a unique solution, when there are no free variables, or (ii) infinitely many solutions, when there is at least one free variable.

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