Linear transformation of doubly-bounded data

Doubly-bounded data, where data values are limited to a fixed range, play an important role in machine learning, where activation functions typically clamp the data to a fixed range, either [-1, 1], or [0, 1]. We assume without any loss of generality that data is limited to $[0 ,\; 1]$. The first operation in a neural network layer is linear transformation. Therefore, linear transformation of doubly-bounded data is central to the study of neural networks.