1
vote
2answers
75 views

Is it true that the whole space is the direct sum of a subspace and its orthogonal space?

Problem The ground field is $K$, $\operatorname{char}K\neq2$. Suppose $W$ is a (maybe infinite dimensional) subspace of a vector space $V$ with a symmetric/symplectic form ...
0
votes
1answer
65 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
0
votes
1answer
79 views

Hyperbolic lattice and its cone

By lattice we mean a finitely generated free abelian group $L$ equipped with an integral non-degenerate symmetric bilinear form $L\times L\rightarrow\Bbb{Z}, \ (x,y)\mapsto x\cdot y$. We call $L$ ...
0
votes
1answer
57 views

What is the geometric interpretation of quadratic forms?

I am trying to make sense of the following condition: Let $w_1, \dots, w_m \in \mathbb{C}^d$ with $\|w_i\| \le 1$ and $\sum_{i = 1}^m \, |\langle u, w_i \rangle |^2 = n$ for some $n \in \mathbb{R}$ ...
0
votes
1answer
92 views

Problem on hyperbolic hyperboloid generated by a rotation

This is the problem: In $\mathbb{E}^3$ we consider the conic $\gamma$ of equations $x=yz-2=0$ , the line $a$ of equations $x=y+z=0$ and the surface $Q$, that is generated by the rotation of $\gamma$ ...
0
votes
1answer
66 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
3
votes
6answers
362 views

Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
2
votes
0answers
110 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...
3
votes
1answer
73 views

Intersection of Sphere and Line in $\mathbb{R}^n$?

This seems to me as a very simple and basic question, though I'm having trouble with it. The Problem Given a sphere $K\in\mathbb{R}^n$ with radius $r\in\mathbb{R}$ and center ...
4
votes
1answer
354 views

Is a general (non-homogeneous) quadratic equation in $\mathbb{R}^3$ an ellipsoid?

This sounds like a simple problem, but I can't get it done. Given the general equation $ax^2 + by^2 + cz^2 + dxy + exz + fyz + gx + hy + iz + j = 0,$ what are the requirements on the coefficients so ...