Lie Groups and Lie Algebras: Winter-2023
HW 4 : Due Wednesday 2/8

1. Constrained dot products I S0 4610S Suppose \(V\subset\mathbb{R}^n\) is a collection of vectors such that \[ 2\> \frac{v\cdot w}{v\cdot v} \in\mathbb{Z} \] for all elements \(v,w\in V\). (\(\mathbb{Z}\) denotes the integers; \(\cdot\) denotes the usual dot product.)
   1. What are the possible angles between \(v\) and \(w\)?
   2. What are the possible ratios of \(|v|\) to \(|w|\)?
      You may assume \(|v|\le|w|\).
2. Constrained dot product II S0 4610S Again assume that \[ 2\> \frac{v\cdot w}{v\cdot v} \in\mathbb{Z} \tag{*} \] for all vectors \(v,w\) in some subset of \(\mathbb{R}^n\).
   1. Choose two vectors \(v,w\in\mathbb{R}^2\) satisfying (*) and such that the angle between them is \(\frac{3\pi}{4}\).
   2. Embedding \(\mathbb{R}^2\) in \(\mathbb{R}^3\), it is straightforward to choose \(u\perp\mathbb{R}^2\) and to show that \(S=\{u,v,w\}\subset\mathbb{R}^3\) satisfies (*). Find some other, linearly independent \(u\), not perpendicular to both \(v\) and \(w\), such that (*) still holds on \(S\).
   3. What are the angles between each pair of vectors in \(S\)? What are the ratios of their magnitudes?
   4. Challenge #1: Can you find a solution satisfying the additional condition that \(v\cdot w\le0\) for all \(v,w\in S\)? What about \(v\cdot w\ge0\)?
   5. Challenge #2: Can you extend your solution to \(\mathbb{R}^4\)? Your four vectors should be linearly independent, and none should be orthogonal to all three of the others.
   6. Challenge #3: Attempt the first two questions assuming the angle is \(\frac{5\pi}6\) instead of \(\frac{3\pi}4\).