Linear transformations

A linear transformation, or linear map, is a function between vector spaces that preserves addition and scalar multiplication. This chapter reads such maps through their geometry, their kernel and image and, in finite dimensions, as matrices once bases are chosen. It assumes the bases, dimension and coordinates of vector spaces.

Lessons

1. Linear transformations: the definition, tests for linearity and why values on a basis determine the whole map.
2. Geometric examples: matrix maps in standard coordinates, read as scalings, reflections, rotations and projections.
3. Kernel, image and rank: the inputs a map sends to zero, the outputs it can produce and their dimensions.
4. Rank-nullity: how nullity and rank add up to the dimension of the domain and what this says about the solutions of $Ax=b$.
5. Matrix representations: the matrix of a map relative to chosen bases and injectivity, surjectivity and invertibility read from that matrix.
6. Seen questions: past exam problems that combine these views in longer arguments.

Where it leads

Once a map has a matrix, the eigenvalues chapter asks whether a different basis can make that matrix diagonal, so that the map acts independently along invariant directions. In Calculus, the kernel of a linear differential operator describes every solution of a homogeneous linear ODE.

Vocabulary checkpoints

• Map and matrix: a linear transformation is a map; its matrix representation depends on the chosen bases.
• Kernel and image: the kernel is a subspace of the domain, while the image is a subspace of the codomain.
• Nullity and rank: nullity and rank are the dimensions of the kernel and the image and they add up to the dimension of the domain.
• Injective and surjective: a linear map is injective when its kernel is trivial and surjective when its image is the whole codomain.
• Linear map and operator: an operator is a linear map from a space to itself and eigenvalues are defined only for operators.