Good day!
Book link: Deep Learning with Pytorch 4
I was going through this chapter Subclassing nn.Module (Chapter 8, Page number 209) from Deep Learning with PyTorch book when I came across the following analogy but not able to fully comprehend:

Second, in one layer, there is not a reduction of output size with regard to input
size: the initial convolution. If we consider a single output pixel as a vector of 32 elements (the channels), it is a linear transformation of 27 elements (as a convolution of
3 channels × 3 × 3 kernel size)—only a moderate increase. In ResNet, the initial convolution generates 64 channels from 147 elements (3 channels × 7 × 7 kernel size). [6]
So the first layer is exceptional in that it greatly increases the overall dimension (as in
channels times pixels) of the data flowing through it, but the mapping for each output pixel considered in isolation still has approximately as many outputs as inputs. [7]

Can someone help with the explanation?

If we consider a single output pixel as a vector of 32 elements (the channels), it is a linear transformation of 27 elements (as a convolution of
3 channels × 3 × 3 kernel size)—only a moderate increase. In ResNet, the initial convolution generates 64 channels from 147 elements (3 channels × 7 × 7 kernel size).


Footnotes:
[6] The dimensions in the pixel-wise linear mapping defined by the first convolution were emphasized by Jeremy
Howard in his fast.ai course (https://www.fast.ai).
[7] Outside of and older than deep learning, projecting into high-dimensional space and then doing conceptually simpler (than linear) machine learning is commonly known as the kernel trick. The initial increase in the
number of channels could be seen as a somewhat similar phenomenon, but striking a different balance
between the cleverness of the embedding and the simplicity of the model working on the embedding.