1 Introduction

2 Linear Transformations

A linear map (also called a linear mapping, linear transformation, vector space homomorphism同态, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication:

• f(u + v) = f(u) + f(v)
• f(au) = af(u)

2.1 Linear transformation is changes of basis

Do you know…

▪what u1  and 𝑢2  are?

▪what a1        and a2  are?

• Think it

2.2 The matrix of a linear transformation

2.2.1 Recall: Matrix Product

右侧更直观：基底的线性组合。把列看成局部坐标系基底的全局表示，x是y在此基底下的局部坐标，得到y的全局向量表示。

2.2.2 The matrix

• Each 2D linear map can be represented by a unique 2×2 matrix

  

• Concatenation of mappings corresponds to multiplication of matrices