It seems there would be some collisions in what the different symbols map to, resulting in loss of information.
For example, say we use the English alphabet, multiply by 2 and take the congruence mod 26. In this case, A and N will both map to the same ciphertext letter, regardless of what the shift value is. Same with B and O, etc. In fact, if the formula is $(ax + b) \mod n$ where $n$ is the size of the plaintext alphabet, then the ciphertext alphabet will have only $1 \over a$ as many symbols as the plaintext alphabet.
Am I misunderstanding something?