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See
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Vectors
- Decision variables, cost vectors, quantities, directions.
-
Matrices
- Constraint matrices, matrix-vector products, standard form.
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Linear Combinations
- Building vectors from other vectors.
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Linear Systems
- Solving systems of equations from active constraints.
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Rank and Linear Independence
- Detecting redundant equations and valid bases.
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Bases and Invertible Matrices
- Basis matrices used in basic feasible solutions and simplex.
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Geometry of Linear Equations
- Lines, planes, intersections, and feasible regions.
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Half-Spaces and Hyperplanes
- The geometry of linear inequalities.
Purpose
Operational research uses optimization models. Linear programming, in particular, is written using vectors and matrices.
The basic form is:
\[\min c^T x\]subject to
\[Ax = b, \qquad x \ge 0\]where:
- $x$ is the vector of decision variables
- $c$ is the vector of objective coefficients
- $A$ is the constraint matrix
- $b$ is the vector of right-hand side values
Why Linear Algebra Matters
Linear algebra gives the language for:
- writing many constraints compactly
- understanding feasible regions geometrically
- identifying vertices and basic feasible solutions
- computing simplex steps
- checking optimality through reduced costs
Core Objects
| Object | Meaning in OR |
|---|---|
| Vector $x$ | decision variables |
| Vector $c$ | costs/profits/objective coefficients |
| Matrix $A$ | coefficients of constraints |
| Vector $b$ | resource limits or requirements |
| Product $Ax$ | left-hand side of all constraints |
| Product $c^T x$ | objective value |
| Basis matrix $B$ | selected independent columns of $A$ |
| Inverse $B^{-1}$ | used to compute basic variables |
Main Mental Model
A linear program is a problem about choosing a vector $x$.
The constraints restrict which vectors are allowed.
The objective chooses the best allowed vector.
In two variables, we can draw the feasible region. In many variables, the same geometry still exists, but we use matrices and algorithms instead of drawings.
Minimal Roadmap
- Learn what vectors represent.
- Learn how matrices represent many equations at once.
- Learn how linear combinations describe dependence and span.
- Learn how rank tells us whether constraints or columns are independent.
- Learn how bases produce basic solutions.
- Use these ideas to understand vertices, feasible regions, and simplex.
Exam checkpoint
Use this topic only as much as needed to support LP algebra: matrices, rank, bases, linear systems, half-spaces, and hyperplanes. Connect every computation back to $Ax=b$, basis matrices, and feasible regions.