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.


A linear transformation (or linear mapping) is a mapping T: VW such that, for each u, v V, and for each cF, T(u + v) = T(u) + T(v), and T(cu) = cT(u).

V is called the domain of the linear transformation T: VW.

W is called the codomain of the linear transformation T: VW.

The identity transformation IV: VV is defined by IV(v) = v for each vV. IV is also denoted by I.

The zero transformation 0: VW is defined by 0(v) = 0W for each vV.

A linear operator is a linear transformation T: VV.


Let T: VW be a linear transformation. The following facts can ...

Get Handbook of Linear Algebra, 2nd Edition now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.