My textbook said "if the message is $A_1A_2...A_n$ and the check digits are $A_{n+1}A_{n+2}...A_{n+j}...$, then each $A_{n+j}$ equals a linear form in $A_1A_2...A_n$ over $\mathbb{Z}_{2}$
What is the meaning of "linear form" in this context? Is it the sum f the form $A_{k1}+A_{k2}+...+A_{ki}$? Also, does the expression "over $\mathbb{Z}_{2}$" mean that the sum is evaluated in modulo 2?
I guess your textbook refers to a linear code. Then it is true that every check digit $A_{n+i}$ can be written as a linear form $\mathbb{Z}_2^n \to \mathbb{Z}_2$ evaluated at $A_1,\ldots,A_n$.
A linear form $f : \mathbb Z_2^n \to \mathbb Z_2$ has the form $$f(x_1,\ldots,x_n) = \lambda_1 x_1 + \lambda_2 x_2 + \ldots + \lambda_n x_n$$ where the $\lambda_i$ are fixed values in $\mathbb Z_2$.