Lecture Note on Linear Algebra 7. Linear Transformations

We will introduce linear transformations here, including its definition and different types of linear transformations. Basic conceptµtransformations(C), linear transformations(5C), standard matrix (IOÝ), onto/surjection(÷), one-to-one/injection(ü §éN), image (), pre–image (), domain (Â), codomain (éN/{Â/m), range of T () 2 Linear Transformations Let A be an m × n matrix. Then for any ~x ∈ Rn, we can obtain another vector ~y = A~x ∈ Rm. Thus, we can create a rule using matrix A which associates each vector in Rn to a unique vector in Rm. For m = n = 1, this association turns out to be a linear function ~y = (a)~x = a~x, which has been discussed extensively in school mathematics. Definition 1 (Transformations(C)). In general, a function T from Rn to Rm is called a transformation (or map) from Rn to Rm, denoted by(P) T: Rn → Rm, ~y = T (~x) (or ~x 7 → T (~x)), where ~y is called the image () of ~x (under T) and ~x is called a pre–image () of ~y, and here Rn is the called the domain (Â) and Rm is called 1 the codomain (éN/{Â/m). In addition, all images T (~x) are called the range of T (¤k|ܧ=¤kT (~x)|Ü¡). Note that the pre–image of ~y may not be unique. In linear algebra, we focus on studying a special type of transformations which has the property called linearity and is defined as follows. Definition 2 (linear transformations(5C)). A transformation T: Rn → Rm is called linear if for any ~v, ~v1, ~v2 ∈ Rn and c ∈ R, 1. T (~v1 + ~v2) = T (~v1) + T (~v2); 2. T (c~v) = cT (~v)