1
vote
1answer
34 views

Maps from $SO(3)$ to $S_1, S_2$, and $S_1 \times S_2$

I am looking for continuous maps between the special orthogonal group of 3x3 matrices and the unit circle, unit sphere, and their product (S1, S2, S2 x S3, respectively). Any hints as to what I should ...
0
votes
2answers
52 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
1
vote
1answer
46 views

Computing the intersections of arbitrary number of planes

Imagine three sets of planes (A, B, and C). The planes in each set are parallel and evenly spaced and so can be defined by a plane equation and a scalar representing the distance between planes. If ...
0
votes
0answers
14 views

Constructing a smoothly varying basis without singularities

I am trying to construct a smoothly varying and a differentiable basis to map a vector in $\mathbf{B}:\mathbb{R}^3 \to \mathbb{R}^3$. Given a vector field $\mathbf{n}(\mathbf{x})$ where $\mathbf{n} = ...
1
vote
1answer
36 views

the “unit speed” anlogue of the evolute of the curve

Given a curve, $\gamma: \mathbb{R} \to \mathbb{R}^2$ define the flow in the normal direction by $\gamma(t) + \epsilon \, \mathbf{n}(t)$. This is different from the evolute which moves at speed ...
2
votes
2answers
39 views

Using a contradiction to show something is not compact

Consider the set of $3$ by $3$ matrices in $\mathbb{R}$ that have nonzero determinant. I want to prove that this is not compact preferably with a contradiction. Attempt: Since its a determinant it ...
0
votes
1answer
27 views

Relations between an affine space and a topological space

What is the relation between an affine space and a topological space? Is one a specialization of the other? Moreover, what do we call a point in geometry: an element of a topological space or an ...
-1
votes
1answer
25 views

Finding a condition to make a map a contracting map

Consider $(\mathbb{C}^n ,||\cdot||_{1})$ where $||x||_{1} = \sum\limits_{i=1}^{n} |x_{i}|$ and $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
0
votes
1answer
46 views

Is the set of all matrices in M(n;R) all of whose eigenvalues satisfy the condition |λ|≤2. compact?

Is the set of all matrices in $M(n; R)$ all of whose eigenvalues satisfy the condition $|λ| ≤ 2.$ compact?
0
votes
1answer
38 views

Hypersphere isometry?

I will denote the $n-$sphere of radius $1$ centered at the origin as $\mathbb{S}^n$, so that $$ \mathbb{S}^n = \{ x \in \mathbb{R}^{n+1}\ : \ \|x\| = 1\}. $$ I am stuck on the following problem...I'm ...
2
votes
3answers
86 views

Positive definite part of a symmetric matrix - or: are the positive definite matrices a retract of the set of symmetric matrices?

$\newcommand{\Sym}{\operatorname{Sym}}$ Denote by $\Sym(n)$ the set of symmetric, real $n\times n$ matrices and let $\iota:\Sym^+(n)\hookrightarrow \Sym(n)$ be the subset of positive definite ...
2
votes
3answers
86 views

topology on $\hom_{\mathbb C}(V,W)$

Let's $V,W$ be finite-dimensional complex vector spaces. How to define topology on $\hom_{\mathbb C}(V,W)$? As I see we can define metric in this $(\dim_{\mathbb C}V\times\dim_{\mathbb ...
1
vote
1answer
28 views

Construct dense and disjoint sets of $\mathbb{R}^m$ so that every element of their Cartesian product has full rank

Or equivalently, can one construct sets $S_1 ,S_2 ,\dots ,S_n \subseteq \mathbb{R}^m$ so that (i) the sets $S_i$ are dense and disjoint; and (ii) if one picks from each set $S_i$ any element $u_i$, ...
1
vote
2answers
52 views

$SO(3)$ with minimal and maximal trace.

Let $O(3)$ be the set of $3 \times 3$ orthogonal matrices. Let $SO(3)$ be a subset of $O(3)$ such that det($A$)=1 for all $A \in SO(3)$. Show that there is a matrix with minimal trace in $SO(3)$ and ...
2
votes
1answer
64 views

Infinite line is closed in $\mathbb{R}^n$

I have been reading the book "Elements of the functional analisys", by Kolmogorov and Fomin. At the chapter of Normed Linear Spaces, page 73 to be precise, the author makes the following definitions: ...
1
vote
2answers
87 views

Questions on positive definite matrices

First, in this discussion, I am only considering real matrices. Second, I have a few questions I am ruminating on related to symmetric matrices. Some of these questions I need someone to say my ...
3
votes
1answer
64 views

Giving an explicit example of a vector that is perpendicular to $v$

Let $v\in\mathbb{R}^3$ be a unit vector. It is possible to show that there exists vectors $\{w_1,w_2\}$ such that $\{v,w_1,w_2\}$ is orthonormal by applying the Gram-Schmidt process, but can we do so ...
1
vote
0answers
16 views

Quotient space of $End(\mathbb{R}^n)$

Let $X=End(\mathbb{R}^n)$ be the set of linear maps $\mathbb{R}^n\rightarrow\mathbb{R}^n$. We endow $X$ with a topology, induced from the metric on $X=\mathbb{R}^{n^2}$. I want to understand the ...
0
votes
1answer
259 views

Convexity of sum and intersection of convex sets

Let $A_i$ be a subset of $\Bbb{R}^m$ which is convex for $i=1,...,n$. How can I prove that the sum of $A_i$ is also convex? I know how to prove it with two sets: Let $x = a_1 + b_1$ and $y = a_2 ...
1
vote
1answer
60 views

For $k \times n$ matrices $X,Y$ of rank $k$, find a characterization of $Y = AX$, $A \in \operatorname{Gl}(k, \mathbb{R})$

Let $F(k,n)$ (with $k < n$) denote the set of $k \times n$ matrices of of rank $k$ (entries in $\mathbb{R}$). We can give $F(k,n)$ a Euclidean topology by identifying it with a subset of ...
3
votes
1answer
353 views

Is this reading path recommended?

Since doing math requires learning it first, I 've chosen a series of books to understand some ''Higher math''(which I want to read over a period of several years),and would like to see some ...
7
votes
2answers
125 views

Is a homeomorphism which maps lines to lines (and fixes zero) necessarily linear?

We know the homeomorphism $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ maps straight lines to straight lines and the zero vector to the zero vector. Is it Linear?? If so, how can we prove it?
0
votes
0answers
50 views

Linear map preserves closedness of a convex set with property on recession cone

Let $\mathbf{E},\mathbf{Y}$ be two euclidean space, $C$ be a non-empty closed convex set in $\mathbf{E}$. The map $A:\mathbf{E}\rightarrow\mathbf{Y}$ is linear, and $N(A)\cap 0^+(C)$ is a linear ...
1
vote
1answer
68 views

Help with a proof of a theorem about convex sets

I'm studying the following theorem: Theorem 1. Let $C$ be a convex set and let $\textbf{y}$ be a point exterior to the closure of $C$. Then there is a vector $\textbf{a}$ such that ...
2
votes
2answers
243 views

Prove that invertible metrices set is an open set in a given space, and the determinant is continuous [duplicate]

Given a matrix $M_{n\times m}$, we can think about it as a vector in $\mathbb{R}^{n\times m}$ (How come?). How can I prove that the set of all the invertible metrices of size $n\times n$ is an open ...
0
votes
1answer
79 views

Set of all matrix of rank $ r $ is open set in $ M_n (\mathbb { R })$

I have no idea how to start it. Actually I have no idea which matrix in $ M_n (\mathbb {R})$ are of rank $ r $. I know all basic result about it. please help me.
6
votes
5answers
209 views

This set of matrices is open

I'm trying to prove that the set of the matrices whose eigenvalues have non-zero real part is an open subset of $M^n$, the set of square matrices with order $n$ which is identify with $\mathbb ...
0
votes
0answers
164 views

What does is mean by differentiate a matrix $E - DB^{-1}C$?

The problem I am trying to solve is: Prove that the set of $m \times n$ is matrices of rank $r$ is a submanifold of $\mathbb{R}^{mn}$ of of codimension $(m - r)(n -r)$. [HINT: Suppose, for ...
1
vote
0answers
38 views

derivative of matrix [duplicate]

Is the nonsingular matrices open? How can I show that every $m \times n$ matrix is in the image of the derivative of an $m \times n$ matrix (how to differentiate it?) Thanks
1
vote
0answers
53 views

GP 1.4.4 An extension of partial converse of preimage theorem.

This is exercise 1.4.4 on Guillemin and Pollack's Differential Topology Suppose that $Z \subset X \subset Y$ are manifolds, and $z \in Z$. Then there exist independent functions $g_1, \dots, g_l$, ...
5
votes
0answers
98 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
3
votes
1answer
65 views

A proof on smooth function that I don't know what to proof.

Here's the question: Suppose $f: U \rightarrow V$ is a smooth map, for $U \subset R^k$ and $V \subset R^\ell$ open sets. That is, all partial derivatives (of all orders) of $f$ exist and are ...
8
votes
2answers
142 views

Two terms that I want to understand: weakest topology and jointly continuous (in the following context).

I was reading an article online, please help me to understand the following lines (in bold letters). - Topological structure: If (V, ‖·‖) is a normed vector space, the norm ‖·‖ induces a metric and ...
2
votes
2answers
76 views

Smooth maps on a manifold lie group

$$ \operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\ \begin{align} &n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\ &n = 2, \operatorname{GL}_n(\mathbb ...
2
votes
1answer
116 views

Derivative of Linear Map

I'm reading Allan Pollack's Differential Topology and got stuck on this argument: In the second paragraph of page 9, section 1.2 he said "Note that if $f:U\to \mathbf{R^m}$ is itself a linear map ...
3
votes
0answers
284 views

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 ?? ...
2
votes
1answer
71 views

Real Projective Space

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 me.
1
vote
1answer
40 views

Locally finite or not

I am tryıng to learn locally finite and can you give an explanation for my green writing please thank you
3
votes
1answer
91 views

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 ...
0
votes
2answers
120 views

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 ...
2
votes
2answers
93 views

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.
2
votes
0answers
128 views

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 ...
1
vote
1answer
124 views

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 ...
2
votes
1answer
87 views

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 ...
2
votes
0answers
45 views

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.
4
votes
1answer
152 views

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 $e_{1},e_{2}$
0
votes
2answers
71 views

Transition formula for 1-forms

I try to solve this question but ı could not.ı am working for my exam.please help me.
2
votes
1answer
64 views

$[T]^{\beta}_{\beta} = \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix}$ provided $T \circ T = T$ [closed]

Let $V$ be a finite-dimensional vector space and let $T:V \rightarrow V$ be a linear map such that $T \circ T = T$. How should one prove that there is a basis $\beta$ of $V$ such that \begin{eqnarray} ...
0
votes
1answer
44 views

Codimensionality: On Cardinality of Linear Equations

How does the codimension of a subspace give the number of linear equations needed to define the subspace?
4
votes
2answers
272 views

Topology of the space of hermitian positive definite matrices

Let $\mathcal{H}_n \mathbb{C}$ be the set of hermitian $n \times n$ complex matrices. This set carries the structure of a vector space over $\mathbb{R}$ under usual addition. It also inherits the ...