for questions about algebraic geometry that focus on affine space. For affine mappings in linear algebra (i.e. linear mappings plus translations), please use the linear-algebra tag or another appropriate tag.

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85 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
2
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1answer
117 views

A question for epigraph and affine function

I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction. First, I quickly recall some definitions: A (closed) halfspace is a ...
2
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1answer
47 views

Find any affine transformation that swaps affine lines

The task is to find any affine transformation that will swap the following two lines: $$L_1:(1,1,1) + span((1,0,2))$$ $$L_2:(1,0,1) + span((1,0,-1))$$ From what I understand there is a number of ...
2
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0answers
66 views

How to describe the variety for lines through the origin

Let $I(x)$ be an ideal generated by the affine variety $X$. $I(X)=\langle xy,yz,zx\rangle$ gives the three coordinate axes through the origin (because it is the intersection of the $3$ planes $xy=0$, $...
2
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0answers
47 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
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0answers
46 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over $\...
2
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2answers
121 views

Prove that $\text{(BE)}\|\text{(JF)}$ using vectors.

Problem Let $\text{ABC}$ be a triangle and let $\text{I}$ , $\text{J}$ and $\text{K}$ be points such that: $\vec{\text{BI}}=\frac{1}{2}\vec{\text{IC}}$, $\vec{\text{AJ}}=2\vec{\text{JB}}$ and $\...
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0answers
199 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps (i.e....
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0answers
61 views

$\varphi$ affine if with $V = \{V_{i}\}_{i=1, .., m}$ affine covering such that $\varphi^{-1}(V_{i})$ affine

I found an exercise of mine, that I solved, but now I am not sure anymore about the details, and some parts of what I wanted to say there. I have the affine open sets $V_{1}, ..., V_{m}$ and $\varphi^...
2
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0answers
54 views

$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$

Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$ Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis. I don't ...
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0answers
119 views

Affine set and linear equation

Prove or disprove the following statement. For any affine set C in R^n, there exists a solution set of linear equation that express C.
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1answer
278 views

Affine Transformation ― correct direction of scale

due to the fact that I am not mathematician I hope the question wont be ejected cause of triviality. But here we go: what is given: In an svg graphic, I have an element on which several are ...
2
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1answer
325 views

Finding the singularities of affine and projective varieties

I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed. I'm not sure if the definition I've been given is ...
2
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1answer
92 views

Domain rotations in the Mellin integral transform

Let's consider the following form of the Mellin integral transform: $$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$ If we scale the domain of the ...
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1answer
76 views

who found that translation in N space is the same as shearing in N+1 space?

According to the wikipedia, Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective ...
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2answers
408 views

n-dimensional affine space that is not isomorphic with the n-dimensional real space

I'm looking for an example of $n$-dimensional affine space that is isomorphic with $\mathbb{R}^n$ as affine space but not with respect to other properties (for example it has different ordering etc.) ...
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4answers
100 views

A function is convex and concave, show that it has the form $f(x)=ax+b$

A function is convex and concave, it is called affine function. That is the function: $$f(tx+(1-t)y)=tf(x)+(1-t)f(y),\, \, t\in (0,1) $$ Force $y=0$(suppose $0$ is in the domain of $f(x)$), we ...
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4answers
114 views

Find the equation of plane containing line described by

Please help me in this really easy task Find the equation of plane containing line described by $x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
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2answers
48 views

Showing a Variety is Rational?

I'm trying to show that the following varieties are rational: $V_1=V(y^2z-x^3)$ and $V_2=V(xyz-x^3-y^3)$. But I can't think of how to show they are birationally equivalent to $\mathbb{A}^n$ or $\...
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2answers
737 views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) on a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
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1answer
59 views

Every line in $\mathbb{R}^2$ can be described…

I came across such a statement: Let $A = \mathbb{R}^2$, $a,b,c \in A$ be points (we treat $\mathbb{R}^2$ as an affine space). Then any line $L \in A$ can be described as $$L = \lbrace s_1 a + ...
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1answer
109 views

Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
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1answer
66 views

The affine line with two points removed

To which affine variety $V$ is $\mathbb{A}^1 \setminus \{0, 1\}$ isomorphic to? What would be the isomorphism in this case? Any help would be appreciated.
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2answers
233 views

Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
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2answers
121 views

Number of parameters to specify an affine transformation in n dimensions

In general, how many parameters does it take to specify an affine transformation in $n$ dimensions, and how does one go about proving this? For example, in 2 dimensions it takes 6 parameters, and in ...
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1answer
850 views

Intersection of affine subspaces is affine

So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again affine, particularly in $\mathbb{R}^...
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1answer
74 views

Why does isomorphism follow from the natural bijection of Hom sets

If X and Y are varieties, and Y is affine, there is a natural bijective mapping of sets $$\operatorname{Hom}(X,Y)\xrightarrow{\sim}\operatorname{Hom}(A(Y),\mathscr O(X))$$ where the left are ...
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1answer
49 views

If $n\geq 2$, why is $k[\mathbb{A}^n\setminus\{p\}]=k[\mathbb{A}^n]$?

Assuming $n\geq 2$, why is the coordinate ring of affine $n$-space over an algebraically closed field $k$ unchanged if we delete a point? That is, if $p\in \mathbb{A}^n$, why does $k[\mathbb{A}^n\...
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1answer
41 views

Prove that this affinity is the identity mapping

Let $\phi : K^n \to K^n$ be an affinity, such that all lines are parallel to their image under $\phi$. Prove that if $\phi$ has two fixed points, then $\phi$ is the identity mapping. My attempt: I ...
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1answer
94 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and $g=x^2-6x+10y^...
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2answers
208 views

Equation of plane — point/vector pedagogy

Suppose we have a point $\mathbf P$ and a vector $\mathbf n$ in plain ordinary 3D space. Here I am deliberately using upper-case letters for points, and lower-case points for vectors, since they are ...
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1answer
180 views

affine set convex set

How to show the following: Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if $\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
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1answer
72 views

Motion in affine geometry

$V$ is a finite euclidean vectorspace and $\sigma:V->V$ is a motion, this means that $d(\sigma(a_i),\sigma(a_j))=d(a_i,a_j)$ for an affine coordinate system $a_0,...,a_n$ I know the following two ...
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1answer
153 views

Detecting the type of singularity with the Jacobian

Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then ...
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5answers
199 views

How to precisely distinguish vectors and points? [duplicate]

Possible Duplicate: Distinction between vectors and points I have a doubt about the distinction between points and vectors. I know there's already a topic about that here in the web site, but ...
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1answer
701 views

What is the relation between complex numbers and transformation matrices?

I read addition and multiplication with complex numbers can be represented as translation and rotation in a 2D plane. I am using this to move around objects on the screen. I have an offset number, ...
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1answer
237 views

This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG =...
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1answer
47 views

affine variety/space vs. toric variety

I think I'm not quite clear on the meaning of a toric variety... Could someone explain the relation/difference between the affine variety/space and the toric variety? I know that affine ...
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1answer
28 views

Affine variety as subset of another affine variety

I am trying to understand why the following statement is true: If $S$ and $S'$ are subsets of $\mathbb{K}[X_1,...,X_n]$ such that $S\subseteq S'$, then $\mathcal{V}(S') \subseteq \mathcal{V}(S)$....
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1answer
37 views

counter example - affine space

In the affine-n-space $\mathbb A^n_k$ (where $k$ is algebraic closed) you can define for an algebraic set $X$: $I(X)=\left \{ f\in k[x_1,x_2,...,x_n] | \forall a \in X \,\,\,f(a)=0))\right \}$ I ...
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1answer
50 views

Proving $\gcd(f_i)=1\Rightarrow \mathbb{A}_\mathbb{C}^n\setminus \{f_i\}$ is not affine

I need to prove the following lemma: Lemma: Let $f_i\in \mathbb{C}[x_1,\dots,x_m]$ s.t. $\gcd(f_1,f_2,\dots,f_n)=1\quad(1<n\le m)$. Prove that the variety $V=\mathbb{A}_\mathbb{C}^m\setminus\{...
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1answer
33 views

Showing space closed under affine combinations is translation of vector space

I'm struggling to reconcile two different definitions of an affine space. The definition in my course notes is: An affine space in $\mathbb{R}^n$ is a non-empty subset closed under affine ...
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1answer
179 views

The maximum of several affine functions is a polyhedral function

A function $f: \mathbb{R}^n \mapsto (-\infty,\infty]$ is polyhedral if its epigraph is a polyhedral, i.e. $$\text{epi}f=\{(x,t)\in \mathbb{R}^{n+1} | \ \ C\left( \begin{matrix} x\\ t \end{matrix} \...
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1answer
33 views

How to compute the normal form of this geometric object?

Given this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$ Maple screenshots: How to put it into the normal form $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
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1answer
58 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
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1answer
23 views

Question about affine isometric action

Recently, I read the book Kazhdan's Property (T). There is a lemma on the page 75 (Lemma 2.2.1) as following: Lemma. Let $\pi$ be an orthogonal representation of $G$ on $H^0$. For a mapping $\alpha: ...
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2answers
294 views

“Averaging” transformation matrices?

I have a question on how best to "average" transformation matrices. Say that I have n number of 4x4 transformation matrices, and I wanted to find a matrix that approximated each one of the n 4x4 ...
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1answer
88 views

Vector bundles on $\mathbb{A}^1_k$ with doubled origin?

One of the most common examples of gluing affine lines is the affine line $\mathbb{A}^1_k$ with doubled origin. Out of curiousity, is there a known classfication of the vector bundles on this space?
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1answer
62 views

Showing affinity of a function - proof help

Let $V$ be the set of sequences whose terms are contained in $\mathbb{R}^n . V$ is the set of functions $x(·) : N → \mathbb{R}^n $ which we denote as $\{x_n\}_n \subset \mathbb{R}^n$. $V$ is a vector ...
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1answer
48 views

The number of $(d-1)$-faces in a $d$-polytope is at least $(d+1)$

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset \mathbb{R}^d$ be a point configuration affinely spanning $\mathbb{R}$ (i.e., $\operatorname{aff}(V) = \mathbb{R}^d)$. Let $H$ be ...