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Linear and Affine Mappings

Mapping $ f:{\bf V}_1 \rightarrow {\bf V}_2$ is linear if, for $ x_0, x_1 \in {\bf V}_1$ ,

$\displaystyle f(\lambda_0x_0 + \lambda_1x_1) = \lambda_0 f(x_0) + \lambda_1 f(x_1).$

Mapping $ f:{\bf A}_1 \rightarrow {\bf A}_2$ is affine if, for $ x_0, x_1 \in {\bf A}_1$ ,

$\displaystyle f(\lambda_0x_0 + \lambda_1x_1) = \lambda_0 f(x_0) + \lambda_1 f(x_1), $ when $\displaystyle \lambda_0 + \lambda_1 = 1.$

Example 2   $ f(x) = 3x$ is a linear mapping from vector space $ {\bf R}^1$ to $ {\bf R}^1$ , since,

$\displaystyle f(\lambda_0x_0 + \lambda_1 x_1)$ $\displaystyle = 3 (\lambda_0x_0 + \lambda_1 x_1)$    
  $\displaystyle = 3\lambda_0x_0 + 3\lambda_0x_1$    
  $\displaystyle = \lambda_0f(x_0) + \lambda_1f(x_1).$    

On the other hand, $ g(x) = 3 x + 2$ is affine from $ {\bf A}^1$ to $ {\bf A}^1$ , since, when $ \lambda_0 + \lambda_1 = 1$ ,

$\displaystyle g(\lambda_0x_0 + \lambda_1 x_1)$ $\displaystyle = 3 (\lambda_0x_0 + \lambda_1 x_1) + 2$    
  $\displaystyle = 3\lambda_0x_0 + 3\lambda_1x_1 + 2 (\lambda_0 + \lambda_1)$    
  $\displaystyle = \lambda_0(3x_0 + 2) + \lambda_1(3x_1 + 2)$    
  $\displaystyle = \lambda_0g(x_0) + \lambda_1g(x_1),$    



Xianming Chen 2006-10-02