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Linear Algebra, Optimisation and Numerical Methods / Vectors, Linear Combinations, and the Dot Product

Matrices and the Column Picture of Ax

Every lesson after this one treats a matrix AA as a machine: feed it a vector x\mathbf{x}, and AxA\mathbf{x} comes out. This note fixes exactly what that machine does, and shows it has two equally valid readings — AxA\mathbf{x} is a linear combination of AA's columns, and it is also a stack of dot products of AA's rows with x\mathbf{x}. The column reading is the one this unit leans on most heavily — from "which b\mathbf{b} can be reached" in the next topic, to the column space in Topic 3, to a linear transformation's matrix in Topic 4 — so this note pins it down carefully before any machinery (elimination, inverses) is built on top of it.
Learning objectives. By the end of this note you should be able to:
  • read AxA\mathbf{x} as a linear combination of the columns of AA, weighted by the entries of x\mathbf{x};
  • read AxA\mathbf{x} equivalently as a vector of dot products between the rows of AA and x\mathbf{x};
  • explain why the two readings must agree;
  • use the column picture to reason about which right-hand sides b\mathbf{b} a system Ax=bA\mathbf{x}=\mathbf{b} can produce.

In this lesson

  1. Matrices
  2. The column picture of Ax
  3. The row picture: Ax as a stack of dot products
  4. The matrix as a linear machine

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