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Linear Algebra, Optimisation and Numerical Methods / Solving Linear Systems and Matrix Algebra

Systems of Linear Equations and the Row Picture

A replicating portfolio, a hedge ratio, an implied factor exposure — each is found by solving a system of linear equations Ax=bA\mathbf{x}=\mathbf{b}. Before building the algorithm that solves such systems, this note asks a more basic question: geometrically, what is a linear system, and when does it have a solution at all? Two complementary pictures answer this — the row picture, where each equation is a hyperplane and the solution is where they all meet, and the column picture from the previous note, where the solution is the combination of columns that reaches b\mathbf{b}. Holding both pictures side by side is what makes the elimination algorithm of the next note feel inevitable rather than mechanical.
Learning objectives. By the end of this note you should be able to:
  • write a linear system in equation form, row-picture (geometric) form, and matrix form Ax=bA\mathbf{x}=\mathbf{b};
  • interpret a 2×22\times2 or 3×33\times3 system as intersecting lines or planes;
  • state, in row-picture language, when a system has no solution, one solution, or infinitely many;
  • restate the same existence/uniqueness question in column-picture language.

In this lesson

  1. A system as intersecting lines
  2. The column picture, side by side
  3. Existence and uniqueness, previewed

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