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.
Questions tagged [affine-geometry]
1202 questions
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How many affine subspaces of dimension $2$ does $\mathbb{Z}_2^3$ have?
How many affine subspaces of dimension $2$ does $\mathbb{Z}_2^3$ have? I think in $\mathbb{Z}_2^3$ any 3 points are affine independent, so any affine combination will determine a uniqe plane. So it will be $C_{2^3}^3 = C_8^3 = 56$ unique planes? Am…
MathLearner
- 750
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Undestandig of definition of an affine subspace
Consider the affine space $\mathbb{R}^{3}$. Then for example the set $ \{(x,y,z):3x-3y+z=0\}$ is an affine subspace. But $ \{(x,y,z):3x-3y+z=2\} $ and $ \{(x,y,z):x^2\} $ are not affine subspaces.
We have defined the affine subspace: $A = w_{0} + V…
user672626
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find the affine transformation of a line
Find the affine transformation that sends the line $ε:3x+2y+4=0$ of $\mathbb{R^2}$ to the line $x=0$
I am having some problems here, and I am getting confused, if anyone could help I would appreciate it.
The first issue that I have is that an…
領域展開
- 2,139
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Example of non-linear affine transformation
Give an example of a non-linear affine transformation.
Is this exercise correct? Since a affine transformation is written as $f(x)=Ax+b$ where $A\in Gl(\mathbb{R},n)$ and $b\in \mathbb{R^n} $ isn't a linear function by definition ?
I thought every…
領域展開
- 2,139
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Dimension of the smallest affine subspace that contains two distinct affine subspaces
Notation : (E,F) denotes the smallest affine subspace of $R^n$ that contains the affine subspaces $E$ and $F$.
Theorem : Let $E$ and $F$ be affine subspaces of $R^n$. Assume that
$E\cap{F}=\emptyset$.…
autodidacti
- 401
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$y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, unknowns $(x_1, \dots, x_p, y)$: Describes a hyperplane of affine space $\mathbb{R}^{p + 1}$?
Let's say we have an equation $y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, where the unknowns are $(x_1, \dots, x_p,y)$. It is said that this describes a hyperplane $H$ of the affine space $\mathbb{R}^{p + 1}$. How is this a hyperplane of the…
The Pointer
- 4,182
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Affine function domain and range
Could somebody what $\mathbb{K}$ signifies in the function definition ?
If it was $\mathbb{R}$ it could mean Real space
spaul
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Showing Affine maps are combinations.
How do you show that an affine map can be written as sum of a linear transformation and a translation by a vector?
user660337
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Affine geometry, coplanar lines
Take two lines given by their equations and I want to know if they are coplanar. If they are either intersecting or parallel, is it enough to conclude that they are coplanar?
Flo
- 113
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An automorphism of an affine space preserves parallelism: question on proof
Definitions:
An automorphism of an affine space is a permutation of the set $\mathcal{P}$ of points that preserves lines and planes (if the dimension it at least 3).
This is the proof given in my book (with exponential notation):
Consider an…
MyWorld
- 2,398
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What is the connection between affine combinations and subtraction in affine spaces
An affine combination is like a linear combination, however for coefficients $a_i$:
$$\forall a_i \in F: \sum_{i=0}^{n} a_i = 1$$
However you can also subtract points from each other in affine space, producing a distance vector in the process, but…
hgiesel
- 1,257
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If $L$ is an affine subspace, $H$ an affine hyperplane, when is $L \cap H$ a hyperplane in $L$?
I'm new to affine spaces, and had a question I'm not quite certain about. Let $E$ be an affine space of dimension $d < \infty$, and $L, H$ affine subspaces of $E$ with $H$ a hyperplane. When is $L \cap H$ a hyperplane of $L$?
I believe this is the…
D_S
- 33,891
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Affine transformations adding matrices
Just a quick question, how does the (-2, 3) matrix become (-4 6) when its added to the other matrix.
Thanks for any help
Regards
Mike
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Find $\mathrm d f_p$ for $f(x,y)=x^2+y^2$
I am doing an exercise about affine map.
Let $f:\mathbb A^2_\mathbb R \to \mathbb R$ be the map such that $f(x,y) = x^2+y^2$ , and $p$ = $(1,0)$. Find d$f_p$.
But I don't know what's the meaning of d$f_p$. Any ideas?
ramsey
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How to create a polygon on an affine space?
If I had an affine space with a basis $\{a_0, a_1, a_2\}$, I could use these points to either create a triangle or select other three points $\{c_0, c_1, c_2\}$ on the plane to create the triangle $\triangle(c_0, c_1, c_2)$ which, I believe, the set…
Atheridis
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