While reading Peter Rhodes' series of articles on the Picastar, he highly recommends the book The Scientist and Engineer's Guide to Digital Signal Processing by Steven W. Smith, Ph.D. and directs the reader to the Analog Devices web site to get a copy. It is no longer available from A.D. but The DSP Guide is freely available as a web site. I am grateful to Steven W. Smith for making this resource available- you can also buy a copy of the book from the site if you like.

Key concepts to understanding Linear systems are Homogeneity, Additivity, Superposition, and Shift-invariance.

The DSP Guide book does an excellent job explaining these concepts- I have a hard time internalizing them without multiple examples. I found a reference published on the NYU web site titled Linear Systems Theory by Professor David Heeger which has some great analogies for internalizing the terms without getting into deeper math (the math is also there in a separate handout PDF).

When a system is defined as linear, it is not needed to define the output by exhaustive analysis. In a linear system with a sine wave input, the output should be at the same frequency as the input with a shift in time and amplitude (scaling).  Homogeneity means the output increases or decreases in direct proportion to the input (not as in P=I²R). Multiple sine inputs are added together to produce the output (additivity). E.G. Jim and Bob talk at the same time- you hear Jim and Bob- not "JimBob".

The principle of Superposition requires both Homogeneity and Additivity in the Linear System. Inputs are composed or synthesized by scaling and combining/adding signals together. When adding (synthesizing) two or more finite quantities you will always get only one result. When decomposing the result you will find an infinite number of results. In DSP, the decision of how to decompose the result does not really matter as we are dealing with a linear system- the output will always be the same as the input, scaled and shifted in time. Use whichever way of decomposing you wish.

Shift Invariance simply means that equal input stimuli result in the equal output results over time. If you hear a car horn blow and ten seconds later hear it again- you should hear the same sound with only the time scale changing (provided nothing else changed).