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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41 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 $G_u$ ...
2
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0answers
35 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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0answers
292 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
2
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2answers
109 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 ...
2
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0answers
168 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 ...
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89 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
2
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0answers
54 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 ...
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1answer
207 views

Prove that for every point in one-sheeted hyperboloid, there exists at least one line which is full contained in it

Please help me with the task: Prove that for every point in one-sheeted hyperboloid, there exist at least one line, which is full contained in it. Firstly, I've noticed that I can transform the ...
2
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0answers
52 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
80 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
178 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
247 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 ...
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1answer
90 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
73 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
348 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
99 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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4answers
70 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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1answer
54 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
91 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
27 views

Affine transformation that sends a conic to itself but does not preserve the focci or the axes [on hold]

So I'm trying to find an affine transformation that sends a conic to itself but does not preserve the foci or the axes. I don't know if this can be done. I'm pretty sure that if it is possible then I ...
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1answer
44 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 ...
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1answer
23 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
84 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 ...
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3answers
306 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
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1answer
81 views

Geometric Deformations

There are geometric transformations such as translation, rotation and uniform scaling (Affine transformations). I am interested in knowing whether there is a separate class of transformations that ...
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2answers
181 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
152 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
69 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
126 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
186 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, ...
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1answer
465 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
222 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
23 views

Dense basic open set contained in dense open subset

For an affine variety $X$ with coordinate ring $A$ it is not hard to see that for $g\in A$ the basic open set (or distinguished open set) $$D(g):=\{ P\in X | g(P)\neq 0\}$$ is dense in $X$ if and only ...
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1answer
34 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
39 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 ...
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1answer
35 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
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1answer
57 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
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1answer
46 views

Application of Desargues' theorem for constructions

I found this interesting document (german) on the internet. On page 8 it says: "Draw a line segment between two given points only using compass and ruler, while the distance between the two points is ...
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1answer
20 views

Alternative characterization of a finite dimensional affine set

As the definition in the S. Boyd's textbook: My question is the following representation: What is the relationship between this representation and the definition above it? EX: sum of elements ...
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1answer
955 views

Affine Transformation matrix for mapping a triangle to another

I'm trying to write a function in Matlab that will give me a matrix T that can be used to multiply points in homogeneous coordinates. This is achieved my mapping a ...
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1answer
29 views

How do we define affine combinations in affine spaces?

Suppose we're given a field $A$ and a vector space $X$ over it. Then an affine space over that is a set $P$ (of "points") equipped with an action $$+ : X \times P \rightarrow P$$ such that $0_X+p ...
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1answer
345 views

Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things: point vector affine space vector space basis frame coordinate system Can you explain me rigorously (in the mathematical ...
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1answer
42 views

Do all the solutions have to be in an affine variety?

An affine variety $V(X)$ is the zero-locus of a set of polynomials. So if the variety is generated by the polynomial $y-x=0$ in $\mathbb{R}^2$, then do all the solutions (i.e., every point satisfying ...
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1answer
106 views

Desargues Theorem and help with its significance

I am still trying to get a hang of drawing the picture. The only idea I get from the theorem is that if two triangles are in perspective from a point, then we the theorem , we also get that the ...
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1answer
255 views

Affine transformation matrix coefficients

In an affine transformation $x \mapsto Ax+b$, $b$ represents the translation; but what does the matrix $A$ represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term ...
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1answer
634 views

Finding an affine combination of a point on a triangle

I have a problem involving affine combinations that I can't figure out how to solve. Given the above picture, write q as an affine combination of u and w. Now, I understand how to write the ...
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1answer
230 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
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1answer
33 views

affine variety of infinitely many polynomials can be represented as an affine variety of its finite subset

Let $f_1,f_2,\cdots$ be an infinite sequence of polynomials in $k[x_1,\cdots,x_n]$ and let $V(f_1,f_2,\cdots)=\{(a_1,\cdots,a_n)\in k^n:f_i(a_1,\cdots,a_n)=0$ for $i=0,1,\cdots\}$. Show that there is ...
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1answer
420 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 ...
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1answer
209 views

Lines projective space

I have a question concerning the answer of Georges Elencwajg in Lines in projective space There he states that the line $\overline {AB}=\mathbb P(\Lambda)\subset \mathbb P^n$ has its points of the ...