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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2
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1answer
35 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
2
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1answer
90 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
2
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1answer
113 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!
2
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1answer
252 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
100 views

Does convexity of all projections imply convexity in higher dimensions?

If I have $n$ 2-dimensional convex "regions" that are the projections along $n$ (independent) dimensions of a n-dimensional compact subspace: Does that imply that the latter is convex? Is the ...
2
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1answer
256 views

On affine spaces, distances, angles, and coordinates

Let's define an affine space as a pair $(A, V)$, where $A$ is a set and $V$ is a vector space, together with a map $V\times A \rightarrow A, \;\; (v, a) \mapsto v + a,$ such that $\forall \, a \in ...
2
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1answer
530 views

Affine subspace iff statement

I cannot find a proof for the following statement, though the statement itself seems common enough: $X$ is an affine subspace of $\mathbb{R}^n \iff \forall a\in X: X - \{a\}$ is a linear subspace. ...
2
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1answer
540 views

Applications of the fundamental theorems of affine and projective geometry.

The fundamental theorem of affine/projective geometry says that a bijection between two finite dimensional spaces that preserves the relation of collinearity is a (semi-) affine/projective ...
2
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1answer
119 views

non-affine functions

it is obvious that if $f$ is an affine function, then $f$ has this property: there exist two function $g$ and $h$ such that $f(t+s)=g(t)+h(s)$ for all $t,s \in\mathbb{R}$. My question is: is there ...
2
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1answer
112 views

Finding a coset

I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$. (This is the ...
2
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1answer
418 views

Turning affine planes into projective planes

How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$? Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point ...
2
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0answers
44 views

Automorphism of $\mathbb{A}^2$ which maps the finite set of points to the finite set of points

Let $\mathrm{k}$ be infinite field. $P_1,\dots,P_n, Q_1,\dots,Q_n \in \mathbb{A}^2$ and $P_i \neq P_j, Q_i \neq Q_j$. I want to find automorphism(in a.g. sense) which maps $P_i$ to $Q_i$. I have tried ...
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0answers
30 views

Dependence of linear algebra theorems of the commutativity of the field.

In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption. However, I was wondering what implications it ...
2
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1answer
62 views

Affine Transformations: Book to Study over the Summer

I've briefly heard of affine transformations in both linear algebra and calculus and I'd like to find a good book on the subject to study over the summer. So what's a good undergrad-level book on ...
2
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0answers
15 views

Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
2
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1answer
139 views

Decomposition of shear matrix into rotation & scaling

How can I decompose the affine transformation: $$ \begin{bmatrix}1&\text{shear}_x\\\text{shear}_y&1\end{bmatrix}$$ into rotation and scaling primitives? $$ ...
2
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1answer
31 views

How to orthogonalize a set of 2x2 matrices?

I have set of 2D affine transformations of images and I need to modify the transformations such way that they become as close to rotations as possible to minimize distortions of images. Let the ...
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0answers
77 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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0answers
55 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
41 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
58 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 $xy=0$, ...
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0answers
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 $G_u$ ...
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0answers
38 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
110 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
173 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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0answers
103 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 ...
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0answers
57 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 ...
2
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1answer
236 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
53 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 ...
2
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0answers
99 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.
2
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1answer
216 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
votes
1answer
288 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
91 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
75 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
374 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
78 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
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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1answer
56 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
121 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
52 views

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

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
27 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
90 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
539 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
88 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
193 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
160 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
70 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
138 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
191 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, ...