  There is strong interest in developing mathematical methods that can be used
to understand complex neural networks used in image analysis. In this paper, we
introduce techniques from Linear Algebra to model neural network layers as maps
between signal spaces. First, we demonstrate how signal spaces can be used to
visualize weight spaces and convolutional layer kernels. We also demonstrate
how residual vector spaces can be used to further visualize information lost at
each layer. Second, we introduce the concept of invertible networks and an
algorithm for computing input images that yield specific outputs. We
demonstrate our approach on two invertible networks and ResNet18.
