Matrix proof - augmented matrix - row reduction - column operation - proof
If we let A be the augmented m x (n + 1) matrix of a system of m linear equations
with n unknowns
Let B be the m x n matrix obtained from A by removing the last
column.
Let C be the matrix in row reduced form obtained from A by elementary
row operations.
Prove the the following statements are equivalent:
(i) The linear equations have no solutions
(ii) If $\displaystyle c_1$,......, $\displaystyle c_{n+1}$ are the columns of A, then $\displaystyle c_{n+1}$ is not a linear combination of
$\displaystyle c_1$,......, $\displaystyle c_n$