All Questions
Tagged with vector-spaces geometry
498
questions
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Outward pointing normal Tetrahedron
For this tetrahedron I need to write down the order of the vertices such that the normal vector points out of the tetrahedron.
For the base DAC, I have drawn the normal vector pointing outwards ...
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What does it mean to multiply rectangular matrices?
I understand matrix multiplication as linear transformation of one or more vectors based on the transformation matrix. This can be visualized when both the matrices are square and are of same ...
2
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Finding a basis of a Subspace
I have a subspace $U = \langle x^2-x+4,x-1,x^2+x \rangle $ of $P_2$ over $\mathbb R$. I need to find a basis of $U$.
We know already that these $3$ vectors span $U$ so we need to check for linear ...
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What does it mean when a system of Linear Equations have more than one solution?
Consider 3 linear equations where one is a linear combination of other two(which are not parallel). Say $a$, $b$ and $a+b$. Now $a+b$ is also a line right? Then how $a$, $b$ and $a+b$ can have more ...
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Is it true that the probability of n hyperplanes with a maximum of K-2 dimensions intersecting in K-dimensional ambient space is 0?
For back-ground, I'm not well-schooled in higher-dimensional geometry, but I'm currently learning statistical data-science in which many methods rely on the properties of higher-dimensional space.
In ...
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What are All the Reflections in Minkowski Space $\mathbb{R}^{1,n}$?
All the literature on reflections in minkowski space, that I have found, have defined ways to reflect about an arbitrary planes or lines and they always add the disclaimer eventually that the plane or ...
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Geometric condition for diagonalisability on image of standard basis via linear transformation in the plane
Let $f:\mathbb R^2 \rightarrow \mathbb R^2$ be a linear map and $v=f(e_1), w=f(e_2)$ where $e_1, e_2$ is the standard basis. Suppose $f$ is invertible (otherwise it is guaranteed to be diagonalisable)....
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Parametrise a Cylinder
I have a cylinder of equation $x^2+y^2=R^2$ where $z$ ranges from $0$ to $h$.
How would I parameterise this? I want to right $r=(R\cos(\theta),R\sin(\theta),)$ but then I can't write $z=z$ because I ...
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Why is the span of a vector a line through the origin? [duplicate]
Why is the span of a vector a line through the origin?
I understand the "formal" definition of a span and its motivation. It is the set of all linear combinations of the vectors given and ...
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Rotation at $\mathbb{R}^n$ [duplicate]
At $\mathbb{R}^2$, we rotate a point (or a vector) $v = \left( v_1 , v_2 \right) \in \mathbb{R}^2$ around a point, by a angle. For example: the rotation of $(1,0)$ around the origin $(0,0)$ by a angle ...
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Does every isomorphism of vector spaces induce an isomorphism of affine spaces?
By reading the first sentence in this article I interpret that, for every projective space, every isomorphism of its underlying vector space gives rise to an isomorphism of projective spaces.
Is this ...
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Basis Confusion
Find a basis of the subspace $U = \langle x^2 −x +4, x −1, x^2 +x \rangle$ of the vector space $P_2$ over $\Bbb R$ of all polynomials of degree at most $2$.
Am I being stupid? I checked for linear ...
- 261
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41
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Norm of a Multivector in $\wedge \mathbb{R}^3$ for calculating the arrea of a polygon.
I am writing some code to explore some interesting things in Geometric Algebra. The general element of my code is multivector $\wedge \mathbb{R}^3$ that forms an 8-dimensional block vector with ...
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2
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126
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Tangent plane to 3 spheres
Given 3 spheres of radius 9 with center at the points $P = (2,1,0)$, $Q = (5,4,0)$ and $R = (3, 1, 2)$. Find the equation, $ax + by + cz = d$, of a plane tangent to the 3 spheres.
I calculated the ...
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When does two lines coincide?
I was going through this book called "A Course in Mathematics for Students of Physics Volume 1 by Paul Bamberg and Shlomo Sternberg". There in a part they said something like this:
...if we ...
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