Handbook of Linear Algebra, 2nd Edition

Chapter 3

Linear Transformations

Francesco Barioli

University of Tennessee at Chattanooga

3.1 Basic Concepts

Let V, W be vector spaces over a field F.

Definitions:

A linear transformation (or linear mapping) is a mapping T: V → W such that, for each u, v ∈ V, and for each c ∈ F, T(u + v) = T(u) + T(v), and T(cu) = cT(u).

V is called the domain of the linear transformation T: V → W.

W is called the codomain of the linear transformation T: V → W.

The identity transformation IV: V → V is defined by IV(v) = v for each v ∈ V. IV is also denoted by I.

The zero transformation 0: V → W is defined by 0(v) = 0W for each v ∈ V.

A linear operator is a linear transformation T: V → V.

Facts:

Let T: V → W be a linear transformation. The following facts can ...