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

Linear Transformations: Definition and Examples

A hedge, a currency conversion, a factor exposure calculation — every one of these is a rule that takes a position vector as input and produces another vector as output, and every one of them happens to be linear: scaling the input scales the output, and combining two inputs combines the outputs. This note gives that rule a name and a precise test, moving from "AA acting on a column vector" (every previous topic) to a linear transformation — a map between any two vector spaces satisfying the same algebraic law, freed from any particular coordinate system. This abstraction is not decoration: it is what makes "the same map, in a better basis" (the next note) and, eventually, diagonalisation possible.
Learning objectives. By the end of this note you should be able to:
  • state the linearity condition and test whether a given map satisfies it;
  • recognise rotation, projection, and scaling as canonical linear transformations;
  • construct the standard matrix of a linear map T:Rn→RmT:\mathbb{R}^n\to\mathbb{R}^m;
  • identify common maps that fail linearity and explain which axiom breaks.

In this lesson

  1. Definition and the linearity test
  2. Canonical examples
  3. The standard matrix of T:Rn→RmT:\mathbb{R}^n\to\mathbb{R}^m
  4. When a map is not linear

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