# Tagged Questions

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### Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
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### Changing local coordinates on a manifold by a diffeomorphism

This is the set up of my problem: Let $M$ be a manifold with local coordinates $x^1,\dots, x^n$. Let $x^1,\dots,x^n,\xi_1,\dots,\xi_n$ denote the induced coordinates on $T^\ast M$. Let $f:M\to M$ be ...
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### Some questions about the proof of the General Linear Group being a manifold.

I understand the idea behind proving that GL(n,$\mathbb{R}$) is a smooth manifold by first using the fact that it is isomorphic to $\mathbb{R}^{n^{2}}$ and using the continuity of the determinant ...
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### Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
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### Parametric equations of manifold

I have am working for a linear algebra test and I realized that I don't know how to solve exercises with linear manifolds even the basic one. W : $x+y-z+u=1$ $2x+u=2$ $z -u=0$ I don't ...
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### Grassmannian, Plucker coordinates

In which books can I find something about the grassmannian and the plucker coordinates ?
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### Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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### Real projective space is Hausdorff

I could not understand the proof of thıs proposition can you help me and give clear explanation.Just can you say how we have (n+1)x2 matrix?? This prove is correct or I need to add something ?? ...
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### Real Projective Space

Corollary $\bf7.15.$ The real projective space $\mathbb{R}P^n$ is second countable. How I can prove this corollary 7.15.I consider that ı can apply the corollary 7.10.But I could not can you help ...
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### Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
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### What is overlop

I want to ask something that what is overlop.My teacher said that For Ex1, Everything is overlop hence it is not locally finite.For example 2,it doesnt overlop.Actually ı could not understand.please ...
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### Topological manifold example

$\theta(x,x^2)=x$ $\Bbb X =${$(x,x^2)| x$ in $\Bbb R$} And V is subset of $\Bbb R$ $dim\Bbb X=1$ My instructor said that this is topological manifold. Why? Please can you explain me? This ...
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### An open cover that is not locally finite

I could not understand that why is not locally finite for example 13.4 can you give me explanation please.
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### Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
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### I did all explanation. Can you just teach me how to calculate this interior product?

Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Show that an orientation form on $S^n$ is $w=\sum _{i=1}^{n+1}(-1)^{i-1}x^i dx^1∧...∧dx^i∧...∧dx^{n+1}$ I ...
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### Manifolds with boundary and definition

Can you help for understanding this definitions in a good way.What is the my problem is that I can not image this definition and proposition in my mind and also ı dont understand the reason that ı ...
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### Orientation-preserving diffeomorphism [duplicate]

Can you help for solving this please. Although I study this subject I could not solve this question please help me ı am willing to learn this question.
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### The open Möbius Band is not orientable

Can you explain my green underlying please.I have confused and ı dont understand why by the continuity of the orientation at the points $(0,0)$ and $(1,0)$ are also $e_{1},e_{2}$
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### Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
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### Why is the cylinder surface on $\Bbb R^3$ orientable?

Why is the cylinder surface on $\Bbb R^3$ orientable? Please can someone explain me clearly?
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### Orientation preserving diffeomorphism.

I am stuck with the question. I guess that I need to write jacobian matrix. But I could not do. Please help me thank you
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### Show that the zero set of $f$ is an orientable submanifold of $\Bbb R^{n+1}$.

Suppose $f(x_1,...,x_{n+1})$ is a$C^∞$ function on $\Bbb R^{n+1}$ with $0$ as a regular value. Show that the zero set of $f$ is an orientable submanifold of $\Bbb R_{n+1}$. In particular, the unit ...
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### How to decide whether F is orientation-preserving or orientation-reversing as a diffeomorphism onto its image.

Let $U$ be the open set $(0,∞)×(0,2π)$ in the $(r,θ)$ -plane $R^2$. We define $F : U ⊂ R^2 → R^2$ by $F (r, θ ) = (r cos θ , r sin θ )$. How to decide whether F is orientation-preserving or ...
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### How do I show that $F^{∗}(dx∧dy∧dz) = ρ^{2} \sin φ dρ∧dφ∧dθ$.

I dont know how to solve. Please help me. I need to understand such types of the question for my exam studyings.
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### Transition formula for 1-forms

$\bf{17.3.}$ Transition formula for $1$-forms Suppose that $(U,x^1,\ldots,x^n)$ and $(V,y^1,\ldots,y^n)$ are two charts on $M$ with nonempty overlap $U\cap V$. Then a $C^\infty \;1$-form $\omega$ ...
122 views

### Problem about differential of a linear map

Please can you tell how to solve this problem clearly? Please solve this explanatorily. Thank you
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### Is $VV^T + D$ a submanifold?

If the positive definite matrix P forms a manifold, is that the subset that {P: P = V V^T + D} where V is a low rank matrix and D is a positive definite matrix a sub-manifold? This idea is ...
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### Finding a subspace whose intersections with other subpaces are trivial.

On p.24 of the John M. Lee's Introduction to Smooth Manifolds (2nd ed.), he constructs the smooth structure of the Grassmannian. And when he tries to show Hausdorff condition, he says that for any 2 ...
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### Exponential Families and Riemannian Symmetric Spaces

Suppose the $f_{X}(x|\theta)$ is a probability density function from an exponential family. Is it true that the Riemannian manifold which has the Fisher information as it's Riemannian metric is a ...
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### Prove that a surface of revolution is a 2dimension manifold

I have a question about surface of revolution. Prove that a surface of revolution is a 2dimension manifold.
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### Pfaffian system of equations

Given a distribution $L^r=\{X_1,...X_r\}$, that is $\{X_1(p),...X_r(p)\}$ linearly independent on each point $p$ on a manifold $M$ of dimension $n$. At a point $p \in M$, the system ...
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### Implicit function theorem : statement and the rank of the matrix

This is perhaps something standard in linear algebra(a subject I am somewhat weak in). So I apologize in advance. I would be grateful if someone can guide me. The statement of implicit function ...
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### Generalization of Grassmann manifold to include translations?

I came across a certain generalization of Grassmann manifolds and was wondering what work if any has been done on it. If you take the space of $n\times p$ real matrices, $n>p$, and define an ...
156 views

### The dimension of linear map

I am reading "Introduction to smooth manifolds" by Lee and one place is very unclear for me: Let $P$ and $Q$ be any complementary subspaces of $V$ (which is an $n$-dimensional real vector space) of ...