# The dual dual linear programme is the original linear programme

Last edited: 2025-12-05

# Statement

Lemma

For a linear programme given by $A, b,$ and $c$ if we take the dual linear programme and then the dual linear programme of that we get back to the original linear programme .

# Proof

Just following the definition the dual linear programme is defined by $-A^T$, $-c$ and $-b$. Then the dual linear programme of that is given by $-(-A^T)^T$, $-(-b)$ and $-(-c)$ which by laws of linear algebra give us $A$, $b$ and $c$ back.