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

Derivation — Proof of the Rank-Nullity Theorem

This sub-note proves Theorem 1 of the previous note, dim⁡(ker⁡T)+dim⁡(im T)=dim⁡V\dim(\ker T)+\dim(\text{im}\,T)=\dim V. The proof strategy — extend a basis of the kernel to a basis of the whole domain, then show the images of the added vectors form a basis of the image — reuses the same basis-extension move as the Steinitz exchange lemma proof, applied here to a linear map rather than to two abstract spanning/independent sets.
Learning objectives. By the end of this note you should be able to:
  • extend a basis of a subspace to a basis of the whole space;
  • show that the images of the extension vectors are linearly independent;
  • show that these images span the image of TT;
  • assemble both facts into the full rank-nullity proof.

In this lesson

  1. Extending a basis of the kernel
  2. The images span the image
  3. The images are independent
  4. Assembling the theorem

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