I am taking the course algebraic number theory ( book J. Neukirch algebraic number theory ) and we reached a point where we are talking about the lattice I know that a lattice is a discrete subgroup of a vector space like for example $\mathbb{R}^2$ a lattice is $\mathbb{Z}^2$ so we can pick any two linearly independent vectors of $\mathbb{Z}^2$ and they will construct a lattice? On $\mathbb{R}^2$ ? And the volume is the area of the parallelogram?
The volume of is defined related to a basis for the subgroup to be the determinant of the basis matrix this in ( An introduction to the Geometry of numbers J.W.S. Cassels) but in Neukirch book the volume is the root of the determinant of the dot product of the basis vectors! So if I pick different linearly independent vectors I will get different volumes $<3,0>,<0,-5>$ the volume is 15 but if I pick the standard basis it is 1. And the book said the volume does not depend on the basis how?
Remember here basis means two elements $u,v$ of the lattice $L$ such that any element of $L$ is obtainable from them as integer linear combination. So two different basis are related by an integer matrix of determinant $\pm1$. If we stick to ordered basis then the sign can be managed.
What you call standard basis may not belong to the Lattice and so there is no ambiguity.