6 Linear Transformations

Having abstracted the objects, we now abstract the maps between them. A linear transformation is a function that respects the two operations a vector space has — and that single requirement turns out to determine a great deal.

The central result is the rank–nullity theorem (Theorem 6.2.8): what a map collapses and what it reaches are not independent, and their dimensions sum to the dimension of the domain. Most of the section’s other conclusions are consequences of that one equation, including the fact that for a map of a finite dimensional space to itself, injective and surjective mean the same thing.

The section ends by connecting back to Section 2. Choosing a basis for the domain and one for the codomain turns any linear transformation into a matrix, and composition of maps into multiplication of matrices. That correspondence is why the two halves of the course are one subject: a statement about matrices and a statement about linear maps are the same statement in different notation.

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