Linear Algebra: Concepts and Methods

We now turn our attention to special types of functions between vector spaces known as linear transformations. We will look at the matrix representations of linear transfor mations between Euclidean vector spaces, and discuss the concept of similarity of matrices. These ideas will then be employed to investigate change of basis and change of coordinates. This material provides the fundamental theoretical underpinning for the technique of diagonalisation, which has many applications, as we shall see later.

7.1 Linear transformations

A function from one vector space V to a vector space W is a rule which assigns to every vector v ∈ V a unique vector w ∈ W. If this function between vector spaces is linear, then it is known as a linear transformation, (or linear mapping or linear function).

Definition 7 .1 (Linear transformation) Suppose that V and W are vector spaces. A function T : V → W is linear if for all u, v ∈ V and all α ∈ ℝ:

1. T(u + v) = T(u) + T(v), and

2. T(αu) = αT(u).

A linear transformation is a linear function between vector spaces.

A linear transformation of a vector space V to itself, T : V → V is often known as a linear operator.