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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Expressing plane and line as affine subspaces

I need to express the following plane, line and the intersection of the two as affine subspaces with a vector $v \in \mathbb{R}^3$ and a subspace $W \subseteq \mathbb{R}^3$. $$P = \{(x_1, x_2, x_3) ...
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65 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 ...
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22 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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172 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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51 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 ...
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37 views

Set of all affine maps as affine space

Given a two vector spaces, the set of all linear maps between them can easily be turned into a vector space again. The same if true for affine maps: Given two affine space $X$ and $Y$, the set ...
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1answer
101 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 ...
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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: ...
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36 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 ...
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113 views

Intersection of two affine varieties

I'm trying to determine the points where two affine varieties defined over $\mathbb{R}$ intersect. Obviously I can just plot them and see what they look like, but I'd have to just look and guess where ...
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70 views

Multi-affine function

Suppose i have a three-variable function f(x1,x2,x3), f:R^3 -> R. If it is linear for x1,x2 and x3 we can say it has the form f(x1,x2,x3) = c1x1 + c2x2 + c3x3 where c1,c2,c3 in R. We can ...
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87 views

Why in the affine space can not we use Grassmann formula?

For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar. For this reason it is not worth the Grassmann ...
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1answer
54 views

Invariants of point sets in an affine space

A distance between a pair of points in an affine space is invariant under translation, rotation and reflection. An angle in a triangle whose corners are tree points is also invariant under scaling. ...
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1answer
77 views

Intersection of affine subspaces of finite codimension in Hilbert space

I'm wondering whether the following assertion is true: Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
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2answers
66 views

Affine geometry about parallelogram

If ABCD is a parallelogram and M,N,P,Q are points on it sides then MNPQ is a paralellogram iff the diagonals intersect at a common point (i.e the diagonals of MNPQ and ABCD intersect at the same ...
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1answer
167 views

Distance between affine space and point

Let $A,A'$ two affine subspaces of a finite Euclidean Vectorspace $V$. Let $p,p'$ two points, such that $d(A,p)=d(A',p')$. $\dim(A)=\dim(A')$ I would like to show that there exists a movement ...
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98 views

distance of an affine subspace to a polytope

I wonder how to prove the following statement. Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
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How to prove we could use mass point geometry to solve all the triangle problem involving ratio between line segment and transversal in a triangle?

what is an easy way to prove that use mass point geometry to solve a problem in the link i provide that is involving cevians in a triangle is same as using the other way in euclidean geometry or ...
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1answer
47 views

Finding the number of solutions to two equations

I have a question: For the following system of linear equations, using Gaussian elimination, decide whether it has at least one solution. If it does, represent the general solution as an affine map ...