Tagged Questions

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Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
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Lines perpendicular to vectors- are they similar triangles?

Please excuse my horrible vector drawing skills. Let us first assume that we have a third vector, called $\Delta V = V_2 - V_1$ Now, these three vectors make a triangle, $V_1, V_2, \Delta V$. Let ...
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Finding a vertex to complete a parallelogram in $\mathbb{R}^3$ and finding a cosine between two vectors.

Trying to solve an exercise regarding vectorial geometry, I have two doubts: For $A,B,C,D \in \mathbb{R}^3$, $$A = (0,1,0)\\ B = (2,2,0) \\ C = (0,0,2) \\ D = (a,b,c)$$ First, determine ...
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Dimension of Hyperplane

Why the dimension of a of N dimensional space hyperplane is N-1? Is there a mathematical ...
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How to prove U•V = |U|•|V|cos(θ), if θ is the angle between |U| and |V|

This is a snippet from my book. How did they get from |U|$^2$ = U • V = |U|•|V| |U|/|V| ?
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Grab major changes along a line

I have a line where there are many points along it. I want to be able to find the major changes in the line and just grab those points rather than grabbing all the points the line has. Here's a quick ...
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Writing a vector as the sum of two other vectors.

Suppose you have 2 vectors $\vec a = (1,1,2)$ and $\vec b = (3,4,-2)$, how would you write $\vec a$ as the sum of 2 vectors $\vec c$ and $\vec d$ where $\vec c$ is in the direction of $\vec b$ and ...
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Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).

I need a help with this question! Show that a square with vertices t, u, v, w has center 1/4 (t+u+v+w).
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How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
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Tensor product, wedge product, Hodge product, dyad, or what?

Suppose I have two vectors $\mathbf{u} = (u_1, u_2, u_3)$ and $\mathbf{v} = (v_1, v_2, v_3)$ in $\mathbb{R}^3$. I can regard $\mathbf{u}$ as a $3 \times 1$ matrix, and $\mathbf{v}$ as a $1 \times 3$ ...
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map colinear triple of points to another triple of points in $\mathbb{R}^2$

Given two triples of pw different colinear points in $\mathbb{R}^2$ so $(x_1,x_2,x_3),(y_1,y_2,y_3) \in (\mathbb{R}^2)^3$. There is a map of the form $T:\mathbb{R}^2\to\mathbb{R}^2,x\mapsto Ax+b$, ...
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Norm, Euclidean Space and Distance

To a complete layman, how would you define the following terms intuitively? $norm$ , $euclidean$ $space$ , and $euclidean$ $distance$ ? Note: I have tagged Linear Algebra and Probability Theory ...
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sweeping edges till they get a given elevation on an oblique plane

I am constructing wireframe model of 3d objects (prisms,..etc.). from a triangular mesh, I have obtained boundary points and fit striaght lines in order to get polygon edges refering to prism ...
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The idea of the transverse to a vector field

I have a quick question. I am independently reading a book on three dimensional geometry and topology. One line has been stumping me. Here is the following paragraph I do not understand: "Let $X$ be a ...
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isolated zero z of X on a star shaped polygon

Consider a polygon that is star shaped with respect to the isolated zero $z$ of $X$. I want to show that the boundary of the polygon can be made transverse to $X$ by jiggling vertices only in the ...
Proof that for two perpendiicular cords intersecting at P $PA+PB+PC+PD=2PO$ using vectors, O being the center.
To prove $\overline{PA}+\overline{PB}+\overline{PC}+\overline{PD}=2\overline{PO}$ I've been able to do \begin{align} \overline{PA}+\overline{PB}+\overline{PC}+\overline{PD} & = ...