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Math Help - Change of bases of a lattice

  1. #1
    Senior Member Pinkk's Avatar
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    Change of bases of a lattice

    Given a basis (a,b) for a lattice L in \mathbb{R}^{2}, show that every other basis for L is of the form (a', b') = (a,b)P, where P is a 2 by 2 integer matrix with determinant plus or minus 1.

    So clearly a' = ra + sb and b' = ta + vb for r,s,t,v \in \mathbb{Z}, but I don't know how to show that the matrix consisting of those integers has determinant plus or minus 1. Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by Pinkk View Post
    Given a basis (a,b) for a lattice L in \mathbb{R}^{2}, show that every other basis for L is of the form (a', b') = (a,b)P, where P is a 2 by 2 integer matrix with determinant plus or minus 1.

    So clearly a' = ra + sb and b' = ta + vb for r,s,t,v \in \mathbb{Z}, but I don't know how to show that the matrix consisting of those integers has determinant plus or minus 1. Any help would be appreciated.

    If you google "integer matrices" or "invertible int. matrices" you can find some interesting stuff, but

    what you want is: an integer matrix is invertible iff its determinant is \pm 1 .

    This is easy to prove and you're invited to try (hint: product rule for determinants).

    Tonio
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  3. #3
    Senior Member Pinkk's Avatar
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    Makes sense. Thank you.
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