Solvability of a System of Linear Equations

Definition: Solution of a System of Linear Equations

A solution of a system of linear equations with $n$ unknowns is any $n$-tuple $(l_1,\cdots ,l_n)$ such that substituting $l_k$ for the $k$-th unknown variable results in all equations of the system being satisfied.

Definition: Solution Space

The solution space of a system of linear equations is the set of all tuples which are solutions of the system.

Definition: Solvability

A system of linear equations is solvable if it has at least one solution.

Theorem: Solvability of a System of Linear Equations

A system of linear equations is solvable if and only if its coefficient matrix $A$ and its augmented matrix $(A\mid \vec{b})$ have the same rank.

$$\mathrm{rank}(A)=\mathrm{rank}(A\mid \vec{b})$$

PROOF

TODO