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Linear Algebra, Optimisation and Numerical Methods / Vector Spaces, Subspaces, and the Four Fundamental Subspaces

The Complete Solution to Ax = b

The nullspace note answered "which combinations produce zero payoff." This note answers the practically more common question: given a target payoff b≠0\mathbf{b}\neq\mathbf{0} (a client's requested exposure, a hedge target), what is the complete set of position vectors achieving it — not just one solution, but every solution? The answer has a strikingly simple structure: one particular solution, plus the entire nullspace layered on top. This single idea — particular plus homogeneous — recurs throughout applied mathematics, from linear systems here to differential equations later in the vault.
Learning objectives. By the end of this note you should be able to:
  • find one particular solution to Ax=bA\mathbf{x}=\mathbf{b} by setting all free variables to zero;
  • write the complete solution as x=xp+xn\mathbf{x} = \mathbf{x}_p + \mathbf{x}_n and justify the decomposition;
  • classify a system's solvability and uniqueness using the rank rr against mm and nn;
  • read the existence/uniqueness table and match a matrix's shape to its solution behaviour.

In this lesson

  1. Particular plus nullspace
  2. Existence and uniqueness by rank

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