Could someone possibly help me in proving this:
Let $A$ be the augmented $m \times (n + 1)$ matrix of a system of m linear equations with $n$ unknowns.
Let $B$ be the $m \times n$ 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 that the following four statements are equivalent:
The linear equations have no solutions.
If $c_1,\ldots , c_{n+1}$ are the columns of $A$, then $c_{n+1}$ is not a linear combination of $c_1,\ldots , c_{n}$.
Rank$(A) >$ Rank$(B)$.
The lowest non-zero row of $C$ is $(0, \ldots , 0, 1)$.
I have been told a good approach would be to show that 1 $\iff$ 2, 2 $\iff$ 3 and 3 $\iff$ 4 however I have got nowhere significant so far.
Thank you for any help you can provide.
Please see the attachment for my working so far.
