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

$\displaystyle c_1$,......, $\displaystyle c_n$