ANN Series 7 - Affine Transformation

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In the context of linear algebra and machine learning, an affine transformation refers to a transformation that preserves points, straight lines, and planes. Vectors are transformed, and translations are allowed. Mathematically, an affine transformation of a vector x can be represented as w.x + b , where: 
- w is a weight vector (matrix in higher dimensions), 
- x is the input vector, 
- b is a bias vector (a translation vector). 

 The operation w.x represents a linear transformation. When you add the bias b, it becomes an affine transformation. The reason w.x is considered part of an affine transformation is because affine transformations include both linear transformations and translations (shifts). 

 Why is it Affine? 

 - Linear Transformation: 

The term w.x alone (without the bias) constitutes a linear transformation. A linear transformation maps vectors to other vectors without changing the origin; it involves scaling and rotating, but not translating (moving the entire space without rotation). 

- Adding the Bias (Translation): 

The addition of b introduces a translation. This step moves the output of the linear transformation w.x by a fixed amount b, which doesn't depend on x. This translation is what makes the entire operation affine, as it allows for shifts in addition to the scaling and rotating performed by the linear part. 

 Properties of Affine Transformations 

 1. Parallelism: 

Affine transformations preserve parallelism. Parallel lines before the transformation remain parallel after it. 

2. Collinearity and Ratios: 

Points that lie on a line continue to do so after an affine transformation, and the ratio of distances along lines is preserved. 

3. Combination of Linear Transformations and Translations: 

They can be viewed as the composition of linear transformations (rotation, scaling, shearing) and translations (shifts). 

 Conclusion 

 The operation w.x by itself represents a linear transformation. However, in the context of affine transformations, it's typically understood to be part of the broader affine operation w.x + b, where the inclusion of the bias b allows for translations, making the entire operation affine. This property is particularly useful in machine learning and computer graphics, where transformations often involve not just rotations and scalings but also shifts to achieve the desired outcome.

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