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: 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.

Facts:

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

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