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