Foundations for Operational Research

The route

1. 01

   Module 1: Vectors, matrices and linear systems

   Vectors, matrices, and systems of linear equations, ending with the two ways of reading the same system. A constraint is a dot product; a program in standard form is a matrix equation; the simplex method is a question about which columns reach the right-hand side.

   • 1
     Vectors and Linear Combinations

     • Given vectors in ${\mathbb{R}}^n$, the learner can add and scale them, form a linear combination, and compute a dot product, saying what each result means geometrically.
     • Given a vector in ${\mathbb{R}}^n$, the learner can compute its norm, produce the unit vector with the same direction, and decide which of two vectors is longer and whether they point the same way, treating length and direction as independent properties.
   • 2
     Matrices as Operators

     • Given a matrix and a vector, the learner can compute $Ax$ and decide whether a product is defined before forming it.
     • Given $Ax$, the learner can express it as a linear combination of the columns of $A$ weighted by the entries of $x$, and use that reading to decide questions about which vectors $Ax$ can reach.
   • 3
     Solving and Characterising Linear Systems

     • Given a system in reduced form, the learner can determine whether it has no solution, exactly one, or infinitely many, justify the verdict from the pivot structure, and for the infinite case describe the solution set rather than one member of it.
     • Given $Ax=b$ with explicit coefficients, the learner can apply row operations that preserve the solution set, reach echelon form without arithmetic error, and say why each operation leaves the solutions unchanged.
   • 4
     The Row and Column Pictures

     • Given a linear system, the learner can state it in both pictures, say what each makes visible, and use whichever one answers the question at hand.
   • 5
     Linear Independence, Rank, and Bases

     • Given a set of vectors or the columns of a matrix, the learner can decide whether they are linearly independent, state the rank, exhibit an explicit dependence relation when one exists, and say whether a given square matrix can serve as a basis.
2. 02

   Module 2: Constrained optimization: variables, objective and constraints

   The shape of the question, before any method: what is being chosen, what is being optimised, what limits the choice, and what an answer consists of.

   • 1
     Constrained Optimization

     • Given a described decision situation in prose, the learner can state the decision variables, the objective and its direction, and the constraints, and can distinguish a decision variable from a quantity determined by the decisions.
   • 2
     From a Described Problem to a Model

     • Given a decision problem described in words, the learner can define decision variables with meanings and units, write the objective with its direction, write one constraint per stated restriction, and state which described quantities are data rather than variables.
   • 3
     Feasible Sets

     • Given a set of restrictions, the learner can determine whether the feasible set is empty, non-empty and bounded, or non-empty and unbounded, and can state that the classification does not depend on the objective function.
   • 4
     Optimal Solutions and Optimal Values

     • Given a solved or solvable optimization problem, the learner can report the optimal solution and the optimal value as distinct objects, can state when the optimal set contains more than one point, and can recognize when no optimal solution exists.
   • 5
     Half-Spaces and Hyperplanes

     • Given a linear inequality and a point, the learner can determine whether the point satisfies it, lies on the bounding hyperplane, or violates it, and can state what the bounding hyperplane is for that inequality.

What finishing means

Finishing this course means you have demonstrated the required skills with the level of support this course currently assesses.

12 required skills. If you reach a lesson without the background it assumes, you are pointed at the prerequisite first, and returned here afterwards.