Algebra I: A Basic Course in Abstract Algebra

Chapter 15

Linear Transformation

In chapter 12 we discussed the linear transformations of ℝ² and ℝ³ We now extend this concept for a general vector space.

15.1 Definitions and Examples

Definition 15.1.   Let V and W be vector spaces over the same field F. A mapping

 

T:V → W

 

is called a linear transformation from V into W if it preserves vector addition and scalar multiplication, i.e.

 

 

Example 15.1.   Let V(F) and W(F) be vector spaces. Then T: V → W defined by T{v) = Ow ∀ u ∈ V where Ow is the zero of W, is a linear transformation. It is called zero transformation as each element is mapped to zero. The transformation I: V → V defined by ...