- #1

- 155

- 11

Let's say we have the Gram-Schmidt Vectors ##b_i^*## and let's say ##d_n^*,...,d_1^*## is the Gram-Schmidt Version of the dual lattice Vectors of ##d_n,...,d_1##. Let further be ##b_1^* = b_1## and ##d_1^*## the projection of ##d_1## on the ##span(d_2,...,d_n)^{\bot} = span(b_1)##. We have the case ##d_1^* \in span(b_1)## and ##\langle d_1^*,b_1 \rangle = \langle d_1 , b_1 \rangle = 1##. Why does this implies that ##d_1^* = b_1 / ||b_1||^2 = b_1^* / ||b_1^*||^2## ?

What I have proven so far is, that ##||d_1^*|| ||b_1^*|| = 1##, but I can not see from the scalar product, that this implies ##d_1^* = b_1^* / ||b_1*||^2##. From what I have proven I can rewrite ##||d_1^*|| = 1/||b_1^*|| = ||b_1^*||/||b_1^*||^2## but I see here no way to deduct that ##d_1^* = b_1^* / ||b_1^*||^2## holds... I suppose you can't just drop the norm here.

If anyone here has any ideas, it would help me a lot!