This indicates that the linear program has been set up to include one or more redundant constraints.

Here, the source of the redundancy is that the set of restrictions {eq}`sumconstraints`.

Here, the source of the redundancy is the structure of restrictions {eq}`sumconstraints`.

Let's explore this further by printing out $A$ and staring at it.

* $(\mathbf{I}_n \otimes \mathbf{1}_m') \operatorname{vec}(X) = q.$

Components of the vectors $u$ and $v$ of **values** are **shadow prices** of the quantities appearing on the right sides of those constraints.

Components of the vectors $u$ and $v$ of per unit **values** are **shadow prices** of the quantities appearing on the right sides of those constraints.

We can write the dual problem as

### Interpretation of dual problem

By **strong duality**, we know that:

By **strong duality** (please see this lecture

{doc}`Linear Programming <lp_intro>`), we know that:

$$

\sum_{i=1}^m \sum_{j=1}^n c_{ij} x_{ij} = \sum_{i=1}^m p_i u_i + \sum_{j=1}^n q_j v_j