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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Examples of applications of the Theorems of Pappus and Ménélaüs.

I'm going to present an exposition about applications of the Theorems of Pappus and Ménélaüs. I need some simple examples of these two theorems. Any links, please?
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1answer
14 views

Does this map define a rational map?

$\phi(x,y)=\frac{y-x^2}{x^2}$ for $\phi:X\to\mathbb{A}^1(\mathbb{C})$ $X$ being a variety $X=V(\langle x^5-x^4+2x^2y-y^2\rangle) \subset \mathbb{A}^2(\mathbb{C})$
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1answer
12 views

Find an affine linear map given two vectors

Find an affine linear map $$\mathbb{Z}_2^5\to\mathbb{Z}_2^5$$ that sends $(0,1,0,0,1)$ to $(1,0,0,1,0)$. So I know that an affine linear map is one of the form $Az+b$ where $b,z\in\mathbb{Z}_2^5$ ...
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5 views

Image and preimage of projection onto affine space

I saw this task some time ago, but I still cannot do it without horrible many calculations. I will be grateful for any hint. Affine subspace $H$ of $\mathbb{R}^4$ given is by equations: $$ ...
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1answer
19 views

Affine contractions from linear contractions?

Let $V$ be a linear space. Consider a contractive linear map $M:V\mapsto V$, $$ \|Mv\|\leq \|v\| \quad \text{for all vectors } v\in V. $$ Now, for some fixed vector $c\in V$, the question is to sort ...
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0answers
10 views

Affine spaces points

I got a question of affine spaces and the question is about prove given L is an affine space. $(L=U+A)$ where $U$ is a vector space and $A$ is a point i.e. $(2,3)$ . So my question is to prove this ...
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22 views

Prove this is an Affine Space

I have a poor knowledge about affine spaces but I know there are ways to prove an affine space using closed affine combinations theory. But this equestion is different. Which is; Let $L = \{(x,y)\in ...
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0answers
10 views

Order of parabolic subgroups of affine Weyl groups

I have a question about computing the order of an arbitrary parabolic subgroup of an affine Weyl group $W_a$. Given a proper subset $I \subset S_a$ associated with the reflections for the fundamental ...
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1answer
26 views

A Deeper Understanding / Interpretation of Homographies

I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form: $$\begin{bmatrix} \vert & \vert ...
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1answer
16 views

Intersection of two circles in projective space

I have checked the existing question Intersection of two circles. and model for intersection of two circles in the complex projective plane - I do not think either of these answers my question. The ...
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1answer
30 views

Affine Subspace Proof

Question: Suppose that $V$ is an $m$-dimensional affine subspace of $\mathbb R^n$, with $m < n$. show that there exist linearly independent vectors $a_1, \dots, a_{n-m}$, and scalars $b_1, \dots ...
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22 views

Show that in finite affine geometry all lines are on the same number of points step by step proof.

i'm wondering if someone could please give me a step by step proof of the following problem Show that in finite affine geometry all lines are on the same number of points
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1answer
19 views

Convert affine coordinates to projective coordinates?

For any rational map represented by $(\frac{x^4+3y}{x^2+1}, \frac{x+1}{y})$ in affine coordinates, write down the corresponding representation $[F_1(X, Y, Z) : F_2(X, Y, Z) : F_3(X, Y, Z)]$ in ...
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1answer
58 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
43 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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0answers
9 views

Lenght of the affine transformation $s_{\varphi_n, \, 1} \cdot \dots \cdot s_{\varphi_1, \, 1}$

Let $\{\varphi_1, \, \dots , \, \varphi_n\}$ be a subset of positive roots of a root system $\varPhi$ and consider the affine reflections $\{s_{\varphi_n, \, 1}, \, \dots , \, s_{\varphi_1, \, 1} \}$ ...
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2answers
36 views

Is the affine curve $y^2=x^4+y^4$ in $\mathbb A^2$ singular?

Is the affine curve $y^2=x^4+y^4$ in $\mathbb A^2$ singular or nonsingular? Find the singularities and show the types of the singularities if the curve is singular. Let $f(x,y)=x^4+y^4-y^2$. ...
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0answers
12 views

Affine application

Consider the affine space $\mathbb{C}^2$ on the field $\mathbb{C}$. How to show that the function $\mathbb{C}^2 \rightarrow \mathbb{C}^2 : (x,y) \mapsto (\bar x, \bar y)$ transforms lines into lines ...
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1answer
41 views

When does an affine manifold inherit a (quotient) group action?

By an affine manifold I mean a real $n$-dimensional manifold $M$ with charts whose transition functions are in the affine group $Aff(\Bbb R^n)$. There are several other equivalent definitions ...
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1answer
11 views

Affine subspaces proposition

I try to prove that a subset of an affine space is an affine space iff it contains the line through every two distinct points of it.
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0answers
10 views

Number of points of an affin line

I want to show that every affin line of $A(V)$ , where $V$ is a vector space on $K$ and $A(V)$ its affin space, has $|K|$ points
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2answers
25 views

Show that some set is an algebraic set

Show that $\{(t,t^2,t^3):t\in k\}$ for a field $k$ is an algebraic set. Just by looking at the points, i see that they are zeros of $F_1(x,y,z)=xy-z$, $F_2(x,y,z)=xz-y^2$ and $F_3(x,y,z)=yz-x^5$. I ...
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2answers
44 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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4answers
183 views

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2.

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2. Here's my attempt at a solution: We know that any rotation in $\mathbb{R^3}$ can be ...
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2answers
37 views

The composition of two homotheties is either a homothety or translation

Let $f$ and $g$ be two homotheties that don't have the same center. What kind of affine transformation is their transformation? I know it can be either a homothety or translation, but I don't know ...
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1answer
27 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 ...
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0answers
13 views

Existence of affine hull of set S

"Obviously, the intersection of an arbitrary collection of affine sets is again affine. Therefore, given any $S \subset R^{n}$ there exists a unique smallest affine set containing $S$ (namely, the ...
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1answer
36 views

How to prove: Every affine set can be expressed as the solution set of a system of linear equations [closed]

How can I prove that every affine set can be expressed as the solution set of a system of linear equations? Please note that set $C \subseteq \textbf{R}^n$ is said to be affine if for any $x_1 , x_2 ...
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1answer
24 views

Counter example of Algebraic sets

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
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1answer
35 views

Check the following sets are algebraic or not?

Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$. $X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called ...
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2answers
107 views

Finding the defining equations for a simple quotient variety

First of all let me note that I have no experience at all with modern algebraic geometry so if at all possible I would appreciate an answer not involving the concept of a scheme. I have however some ...
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1answer
36 views

Flexible affine variety

Let X be an affine variety. Aut(X) is automorphisms group of X. SAut(X) is a subgroup spanned by all unipotent(image of additive group of field K) subgruops. X is called flexible if SAut(X) act ...
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0answers
16 views

How do projections in 3D with homogeneous coordinates work?

Affine 3D transformations can be expressed in homogeneous coordinates by a matrix $M \in \mathbb{R}^{4 \times 4}$. This means we have 16 parameters to calculate. The first thing I asked myself is how ...
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5answers
67 views

Prove 3 vectors are collinear

I am asked to prove A(2,4), B(8,6), C(11,7) are collinear using vectors. I can work AB by subtracting A from B and BC by subtracting B from C in vector form. I can say that BC = 2AB. But I don't ...
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1answer
36 views

Show that a set is an affine subset

Let $V$ be a vector space over $\mathbb{F}$ and $S \subseteq V$ a nonempty set. Theorem: $S$ is an affine subset of $V$ if and only if $$ \forall u,v \in S, \forall \lambda \in \mathbb{F}, ...
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0answers
16 views

Dimesion of an affine variety- solution verification

I have affine variety $V=\{(t,t^2,t^3)|t\in\mathbb{Q}\}$ and if I'm right I have $I(V)=(y-x^2,z-x^3)$ and coordinate ring is $\mathbb{Q}[V]\cong \mathbb{Q[t]}$ (I set $x=t$). I have this definiton : ...
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1answer
37 views

What's the intuition behind the definition of the tangent space of $\Bbb R^2$?

I'm reading a book on differential forms and on page one it defines the tangent space to $\Bbb R^n$. In what follows I've translated the statements into two dimensions for simplicity. Let $p$ be a ...
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0answers
8 views

Find the homothetic transformation

In $\mathbb{R^3}$: Find the homothety $\Phi$, such that the following transformations are possible: $$\Phi(P)=\Phi(1,0,-1)= (2,5,0)$$ and $$\Phi(Q)=\Phi(0,1,2)= (0,5,2)$$
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32 views

affine transformations, strategy for finding invariant straight lines

At first lets introduce some notation. $\mathcal{A}^n$ is a $n-$dimensional affine space and $V$ is its associated vector space. For any affine subspace of $\mathcal{M}$, its associated vector space ...
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0answers
14 views

How to go from Affine to a Non-linear transformation

If you were able to take two sets of matrices and transform one to attempt to match the other using an affine method such as Least Squares, how could you replicate this process better using a ...
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0answers
7 views

Transforming a vector configuration to an affine point configuration

I am reading the first chapter of Oriented Matroids by Bjorner et. al, in which they consider the set of vectors $\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_6$ given as the columns of the following ...
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1answer
29 views

Affine variety of geometric progressions.

Let $M$ be affine variety $M \subset \mathbb A^n$. $z\in M \Leftrightarrow z = (x, xy, xy^2, ... , xy^{n-1}),where\hspace{2mm} x,y \in \mathbb C, x \neq 0$. I need to find the equations $f_1, ..., ...
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3answers
87 views

How to gain an intuition of the affine function's definition?

Here is the definition of Affine Functions according to Stephan Boyd (EE263 Stanford) : 1- I believe linearity is more restrictive property of a function than being affine since it requires $f(0) = ...
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1answer
26 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
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0answers
37 views

Why is half space not affine?

I've read that half spaces are convex but not affine. I'm trying to understand this geometrically. Does it mean that if I connect any 2 points in the half space, it may result in a line that extends ...
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0answers
8 views

There are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} \cap l_{3} $ is fixed point.

$l_{1}, l_{2}, l_{3} $ are 3 pairwise orthogonal lines in $\mathbb{E_3}$ Prove that there are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} ...
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22 views

Theorem of Pappus (analytical proof)

Let $x$, $y$, $z$, be points on a line $L$, and let $x'$, $y'$, $z'$ be points on a line $L'$. Assume $xy'\parallel x'y$ and $y'z\parallel yz'$. Show that $xz'\parallel x'z$. I want to prove this ...
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1answer
26 views

Find a plane equation from a line and a point

Consider the point $B(1,0,1)$ and the line $R=(1,1,1) + \alpha[1,1,-1]$. Find the equation of a plane that passes through $B$ and contains $R$. What I tried doing was simply setting two different ...
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1answer
22 views

Relative postion of a plane and a hyperplane in $\mathbb{R^4}$

I know what happens in $\mathbb{R^2}$ and $\mathbb R^3$. In $\mathbb{R^2}$ , two lines either intersect in a point or they are parallel. In $\mathbb{R^3}$, two lines (or a line and a plane) can ...
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2answers
24 views

Find a plane perpendicular to the intersection of two affine subspaces

Consider the two following affine subspaces of $\mathbb{R^3} $: $$S=\{x,y,z)\mid 2x+y+z=1\}$$ and $$T=\{(x,y,z)\mid x-y+2z=0\} $$ Find the plane $H$ perpendicular to the intersección of $S$ and $T$, ...