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

The Four Fundamental Subspaces

Every matrix carries exactly four subspaces attached to it, and this note names all four at once, gives their dimensions, and previews the orthogonality between them that Symmetric Matrices later exploits. Strang calls this picture "the heart of linear algebra" — it is the single organising diagram this topic has been building toward: the column space (Topic 1) and nullspace already introduced now sit alongside their transposed counterparts, and their four dimensions satisfy the exact accounting pattern that becomes the rank-nullity theorem once restated for a general linear map in the next topic.
Learning objectives. By the end of this note you should be able to:
  • name the four fundamental subspaces of any m×nm\times n matrix and state which space (Rn\mathbb{R}^n or Rm\mathbb{R}^m) each lives in;
  • state each subspace's dimension in terms of the rank rr;
  • find a basis for each of the four subspaces from a matrix's row echelon form;
  • read the finance meaning of each subspace for a factor-loading or payoff matrix.

In this lesson

  1. The four subspaces
  2. Finding bases for all four

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