Linearity in Pricing and Duality, UMP EMP I have read the statement below and don't quite understand what it means. This is probably because I don't have full understanding of duality with support function in math, but just to foster understanding... The statement is:


  "The utility maximization problem(UMP) has a non-linear utility objective and a linear price constraint. The expenditure minimization problem is exactly the other way around: a linear price objective and a non-linear utility constraint. This is at the heart of the idea of the meaning of dual - to recast the problem with the objective as a constraint and the constraint as the objective. An important observation to make is linearity in pricing in them. This linearity, either in objective or constraint, is the cornerstone in enabling us to use the duality theorem."
Denote a bundle of goods $\mathbf{x}=(x_1,...,x_n)$ and a price vector $\mathbf{p}=(p_1,...,p_n)$. The cost of any bundle is the scalar product $\mathbf{p}\cdot\mathbf{x}$.

Consider first the UMP, which consists in finding the bundle $\mathbf{x}$ that maximizes utility $u(\mathbf{x})$ subject to the budget constraint that the bundle of good costs less than an income $R$:
$$\underset{\mathbf{x}}{\max}u(\mathbf{x}) \text{   s.t.   } \mathbf{p}\cdot\mathbf{x}\leq R.$$

Consider now the EMP, which consists in finding the bundle $\mathbf{x}$ that minimizes expenditures $\mathbf{p}\cdot\mathbf{x}$ subject to achieving a minimal level of utility $\underline{u}$:
$$\underset{\mathbf{x}}{\min}\mathbf{p}\cdot\mathbf{x} \text{   s.t.   } u(\mathbf{x})\geq \underline{u}.$$

These problems are dual in the sense that you can relate the solution of the first one to the solution of the second one. The solution of the UMP gives you the Marshallian demands, $\mathbf{x}^M(\mathbf{p},R)$, as the function of the prices and the income. The solution of the EMP gives you the Hicksian demands, $\mathbf{x}^H(\mathbf{p},\underline{u})$, as the function of the prices and the minimal level of utility. 
The link between these two functions is heavily studied in microeconomics, it is rich in many economic intuitions and useful mathematical relationships. 

Duality enables you to find one solution once you know the other. For instance suppose we know the function $\mathbf{x}^M(\mathbf{p},R)$ but we do not know the function $\mathbf{x}^H(\mathbf{p},\underline{u})$, how can we find $\mathbf{x}^H(\mathbf{p},\underline{u})$?
First, denote the indirect utility of the first problem by $v(\mathbf{p},R)=u(\mathbf{x}^M(\mathbf{p},R))$. Second define the income $\underline{R}$ as the solution of $v(\mathbf{p},\underline{R})=\underline{u}$ (the existence of $\underline{R}$ comes from the linearity property that is emphasized in the last sention of the citation). $\underline{R}$ is the minimum amount that is required to achieve utility $\underline{u}$. I will not show, but you can understand intuitively that we have:
$$\mathbf{x}^H(\mathbf{p},\underline{u})=\mathbf{x}^M(\mathbf{p},\underline{R})$$

Lastly, note that we should write $\underline{R}$ as a function of $\mathbf{p}$ and $\underline{u}$ to be rigorous.