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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1answer
73 views

Show existence of linear transformation from subset to subspace embedded in $\mathbb{F}_2^n$

Assume I have a subset $X$ (not necessarily a subspace) of $\mathbb{F}_2^n$, of size $\leq 2^{n-1}$. It seems likely to me that there should exist a bijective linear transformation taking $X$ to a ...
1
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1answer
76 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 ...
0
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0answers
14 views

How is affine space analogue for lattices called?

Lattices are so like vector spaces that it seems natural to have an affine space construction for them. Unfortunately I could not find how such a construction is called. Could you please help me? ...
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1answer
31 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
0
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1answer
38 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
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0answers
52 views

Linear algebra, affine space, and floor function

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for ...
0
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1answer
48 views

Direction of traslation of affine movement

I have a doubt about this. We have an affine isometry of an affine space $X$ of dimension 3. Now, we know it's the composition of some movement (reflection, rotation, etc), with a traslation, and we ...
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1answer
45 views

Why all norms define the same relative interior?(Convex Optimization, Stephan Boyd)

When I was reading 'Convex Optimization', Stephan Boyd, I was stopped by Example 2.2. Before Example 2.2 is started, following definition is coming. If the affine dimension of a set ...
2
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0answers
34 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 ...
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1answer
46 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
3
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1answer
198 views

Prove, in this figure, that $EFGH$ is a parallelogram

In the following figure, $ABCD$ is a parallelogram, and $O$ is any point. Parallelograms $OAEB, OBFC, OCGD, ODHA$ are completed. Prove that $EFGH$ is a parallelogram. We can obtain a fairly trivial ...
2
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0answers
258 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 ...
3
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2answers
172 views

How to determine the period of a binary matrix?

As a part of my research project, I encounter a difficulty to compute the period of a binary cat matrix $C$, which is a square matrix of dimension $n$ with the following properties $|\rm{det}(C)|=1$ ...
0
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1answer
202 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 ...
0
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1answer
73 views

Affine stratification of Grassmannian $\mathbb{G}(1,\mathbb{P}^3)$

Let $G=\mathbb{G}(1,\mathbb{P})$ be the Grassmannian variety of lines in $\mathbb{P}^3$. I have to do an affine stratification of $G$. In order to do this we consider the flag $\mathcal{F}$ of the ...
0
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1answer
25 views

Find $\vec{QB}$ in terms of $\mathbf c$

I've managed to work out “$\vec{AM}$ in terms of $\mathbf a$ and $\mathbf b$” to be $3\mathbf a+\mathbf b$. But how can I work out “$\vec{QB}$ in terms of $\mathbf c$”?
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2answers
137 views

Get affine transformation matrix from two positions of the object

I have an object in 3d space which is represented by the set of vertices. Then I scale this object, rotate it and translate. After these operations I get the second set of vertices with new ...
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0answers
186 views

Prove, using vectors, that this quadrilateral is a rhombus

Consider the following quadrilateral $ABCD$, with $E, F, G, H$ as the midpoint of $AD, DC, CB, BA$ respectively such that $\Delta ECH$ and $\Delta AGF$ are equilateral. Prove that $ABCD$ is a ...
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1answer
518 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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2answers
31 views

How do you rotate a vector by $90^{\circ}$?

Consider a vector $\vec{OA}$. How will I rotate this vector by $90^{\circ}$ and represent in algebraically?
0
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3answers
56 views

Prove, using affine geometry, that in this figure $\Delta DEF$ is always equilateral

Consider the following figure: $\Delta DAC, \Delta CEB, \Delta AFB$ are isosceles. $\angle ADC = \angle CEB = \angle AFB = 120^{\circ}$. Prove that $\Delta DEF$ is equilateral. Now, there is a ...
0
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1answer
51 views

Prove for any four points: $|AB|^2 + |CD|^2 -|BC|^2 - |AD|^2 = 2\cdot \vec{AB}\cdot \vec{DB}$

Let $A, B, C, D$ be four points in space. Prove $$|AB|^2 + |CD|^2 -|BC|^2 - |AD|^2 = 2\cdot \vec{AC}\cdot \vec{DB}$$ Clearly, $$AB = B-A$$ $$CD = D-C$$ $$AD = D-A$$ If I directly substitute the ...
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2answers
52 views

Determining the rotation shape

Consider a large number of points distributed on the circumference of a circle with radius r. If I rotate each point with a randomly chosen Euler angle around a randomly chosen coordinate inside this ...
0
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1answer
79 views

alternative definition of Affine map

Let $f:X\longrightarrow Y$ be a function on real vector spaces (note that $X,Y$ have arbitrary dimensions). If $T(x)=f(x)-f(0)$ is linear, $f$ is called an affine map. Prove that $f$ is affine if ...
1
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1answer
183 views

Affine group, semi-direct product and linear transformations

According to wikipedia the Affine group is the semi-direct product of a vector space $V$ and the general linear group $GL(V)$. Here is the definition of the semi-direct product in terms of matrices ...
0
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1answer
111 views

Proving a vector equality in a triangle without using Thales' theorem.

Problem Let $\text{ABC}$ be a triangle, and $\text{M}$ and $\text{N}$ are points where: $\vec{\text{AM}}=\frac{1}{3}\vec{\text{AB}}$ and $\vec{\text{AN}}=\frac{1}{2}\vec{\text{AB}}$ and ...
2
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2answers
106 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
162 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 ...
2
votes
1answer
91 views

Affine hull of the intersection of two convex sets

Is it true that the affine hull of the intersection of two convex sets is the intersection of the affine hulls of these sets? Where the intersection of the two convex sets is non empty? Many thanks!
4
votes
1answer
90 views

Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$

I don't know how to prove the following: Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$, where $A \in GL_2(\mathbb{R})$, $b$ is a fixed vector in ...
0
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1answer
73 views

Problem with vector calculation.

Problem Let $\text{ABC}$ be a triangle and let $\text{A'}$ , $\text{B'}$ and $\text{C'}$ be respectively the center of $\text{[BC]}$ , $\text{[AC]}$ and $\text{[AB]}$. Prove that: ...
3
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1answer
582 views

Affine sets and affine hull

Mathematically an affine hull can be expressed as $ Aff[C] = \{\theta_1x_1 + \theta_2x_2 .... \theta_nx_n| x_i \in C \ \ \sum_{i=1}^{n}\theta_i = 1 \}$ Intuitively can anyone explain what this ...
4
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0answers
105 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
0
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1answer
37 views

Affine coordinates of a line

Can you help me figuring out how to solve the next problem? If the points M and N have affine coordinates $(m_1,m_2,m_3)$ and $(n_1,n_2,n_3)$ with respect to some points A,B,C, then the points X of ...
3
votes
1answer
126 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
0
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1answer
35 views

Effect of Moving within the Feasible Region

$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a concave function with local maximum at $\mathbf{x}^*$ in a convex, closed feasible set $\mathcal{F}\subset\mathbb{R}^n$. Now consider a suboptimal point ...
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0answers
42 views

affine translation in direction of a vector

Suppose I have a line segment in 3D-space, having end-points $(a,b)$. I want to translate this segment by $w$ units in the direction specified by 3 angles $\alpha,\beta,\gamma$ with respect to ...
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1answer
74 views

The graph of a regular function is an algebraic set, and intersection of hypersurfaces is finite?

i have some problems with these exercises, can you give me a hint? Let $f:\mathbb A^n_k\rightarrow\mathbb A^m_k$ be a regular function. If $X\subset\mathbb A^n_k$ is an algebraic set, show that the ...
4
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0answers
92 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
2
votes
1answer
275 views

Best book to learn Affine Geometry?

I'm going to learn Affine plane as well as affine Geometry. Unfortunately, my text book (not in English) is not good at all, so please recommend some book you think it's good for self-learning (and ...
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0answers
121 views

Number of vector and affine subspaces of dimension $ k$ of $E$ over $\mathbb{F_q}$

Problem (comments after): Let $\mathbb{F_q}$ be a finite field of cardinal $q$ and $\mathcal{E}$ an affine espace of dimension $n$ directed by the vector space $E$. Show that: ...
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1answer
202 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 ...
2
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1answer
93 views

Whitney umbrella birational to $\mathbb{A}^2$ but not isomorphic

Define the Whitney umbrella as the affine surface $V(z^2 - yx^2) \subset \mathbb{A}^3$. I've come across an exercise that asks me to show that this surface is birational, but not isomorphic, to ...
1
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1answer
163 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
2
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0answers
73 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 ...
3
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1answer
106 views

Prove that $\mathbb{A}^1 - \{p_1, \dots, p_n\}$ and $\mathbb{A}^1 - \{q_1, \dots, q_m\}$ are not isomorphic for $n \neq m$

I want to prove that $\mathbb{A}^1 - \{p_1, \dots, p_n\}$ and $\mathbb{A}^1 - \{q_1, \dots, q_m\}$ are not isomorphic for $n \neq m$, where the $p$'s and $q$'s are points. This is one of those ...
3
votes
1answer
164 views

Complement of a line in $\mathbb{A}^2$ as an algebraic variety

I just started reading notes from an algebraic geometry course, and I'm curious about whether the complement of a line in $\mathbb{A}^2$ is always an algebraic variety. If so, what does it's ...
3
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0answers
106 views

Smallest $\sigma$-algebra of $\Bbb A^n$ containing all affine algebraic subsets.

Let $k=\overline k$. What is the smallest $\sigma$-algebra $\Sigma$ containing all affine algebraic subsets? I am interested in the analogous question for $\operatorname{Spec} k[x_1,\dots,x_n]$, but ...
2
votes
1answer
92 views

Affine space $A^n$ and definition of difference.

I'm not sure if this question would be more appropriate in Physics.SE, if so let me know. I need help in understanding this quote from "Arnold - Mathematical Methods in Classical Mechanics" (This is ...
1
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1answer
69 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 ...