Linear cryptanalysis and number of linear approximations I try to understand the linear cryptanalysis. Now I am reading as a secondary reference the book "Cryptanalysis of Block Ciphers:
A Survey" by Standaert , Piret and Quisquater. In the first step, the authors are saying that there are $(2^n-1)\times(2^m-1)$ linear approximations for a $n$-input vector and a $m$-output vector.

Since a linear approximation is a linear map between $\mathbb{F}_2^n\rightarrow\mathbb{F}_2^m$, should the number of linear approximation not be $2^n\times 2^m$? Or do I miss a part?

I am grateful every help.

Sincerely,
Hypertrooper
It most likely assumes both $a$ and $b$ are nonzero in
$$
a \cdot x + b\cdot y.
$$
If either are zero, the overall inear approximation will be insensitive to changes in plaintext (respectively ciphertext) making the overal linear expresion useless to mount a linear attack on a multiround cipher, since the idea is to extract subkey bits by using enough known ciphertext plaintext pairs.