Duality of a linear programming problem in matrix form?

Question

trying to find the duality of the LP problem in matrix/vector form:

min c₁^Tx₁ + c₂^Tx₂
s.t. A₁x₁ + A₂x₂ = b
x₁>= 0

I get that the duality of like

min c^Tx
s.t. Ax = b
x>=0

would be

max b^Tu
s.t. A^Tu <= c

but I'm not really sure how the addition of 2 different matrices would affect the constraints or how to visualize this problem? I think the objective would still be max b^Tu but I am not sure?
Thanks.

Answer 1

I assume that $x_1$ and $x_2$ are single variables. Then the dual program is

$\texttt{max} \ \ b^Tu$

$A_1^T u\leq c_1$

$A_2^T u=c_2$

The number of variables ($u_i$) depends on the dimension of vector $\mathbf b$.