# Is there any concept like 'applying affine transformation on multiple inputs'?

Affine transformation on $$X$$ is a transformation of the following form

$$Y = wX + b$$

In general, $$w, X, Y$$ and $$b$$ tensors.

We generally call tensor $$X$$ as an input to affine transformation or the tensor which we want to transform. We call $$w, b$$ as weight and bias tensors respectively. We call $$Y$$ as output tensor after transformation. Every layer of multi layer percetron contains an affine transformation.

Suppose I have two types of inputs, say $$X_1, X_2$$. Now, I want to apply affine transformation on them using other.

Consider the following

#1: Combining using individual affine transformations

$$Y_1 = w_1X_1 + b_1$$ $$Y_2 = w_2X_2 + b_2$$ $$Y_1Y_2 = w_1w_2X_1X_2 + w_1b_2X_1 +w_2b_1X_2 + b_1b_2$$

#2 multiplying them and applying affine

$$Y = wX_1X_2 + b$$

#3 concatenating them and applying affine

$$Y = w (X_1, X_2) + b$$

Does anyone of the above eligible to call affine transformation in terms of $$X$$ and $$Y$$ (not $$XY$$)? If not, is it true that there is nothing like affine transformation on two inputs taken together?