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Linear Algebra, Optimisation and Numerical Methods / Linear Transformations, Matrix Representation, and Rank-Nullity

Kernel, Image, and the Rank-Nullity Theorem

This note delivers the unit's first deep structural theorem, promised in the conceptual spine of the unit: the rank-nullity theorem, dim⁡(ker⁡T)+dim⁡(im T)=dim⁡V\dim(\ker T)+\dim(\text{im}\,T) = \dim V. It is deep in a precise sense — it is not obvious from the definitions alone, it holds for every linear map between finite-dimensional spaces with no exceptions, and it is the abstract statement behind a pattern already observed twice: the nullspace and column space dimensions summing to nn, and the four fundamental subspaces' dimension table. This note renames those objects for a general linear map and states the theorem precisely; the derivation sub-note carries the full proof.
Learning objectives. By the end of this note you should be able to:
  • define the kernel and image of a linear map and verify each is a subspace;
  • state the rank-nullity theorem and connect it to the nullspace/column-space dimension pattern of Topic 3;
  • use rank-nullity to determine injectivity and surjectivity from dimension counts alone;
  • apply the theorem to a concrete linear map and interpret the result.

In this lesson

  1. Kernel and image
  2. The rank-nullity theorem
  3. Injectivity, surjectivity, and rank-nullity

Before you start

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