# Tagged Questions

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### Intuitive idea of immersions?

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, ...
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### Projection from triangle to $\mathbb{R}^2$.

I constructed the $2$-simplex as follows, $$\triangle^2= pe_1+qe_2+re_3 \hspace{4mm} p,q,r \in \mathbb{R}$$ I want to project this triangle down to $\mathbb{R}^2$, that is so I can write ...
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### P-norm Unit Ball

Proof that for $0<p<1$, $p\in \Bbb{R}$ $$\|(x,y)\|_p=(|x|^p+|y|^p)^{\frac{1}{p}}$$ doesn't define a norm in $\Bbb{R}^2$. However, $$d_p((x_1,x_2),(y_1,y_2))=\sum_{i=1}^2|x_i-y_i|^p$$ defines a ...
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### Is the set of invertible upper triangular matrices open in $GL_n(\mathbb R)$? Is it open in the set of all upper triangular matrices?

I think the answer to the second question is yes, but can't quite prove this. I've no idea about the first part. I've done a few exercises of this kind but all have used the continuity of the ...
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### Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
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### How to prove the equivalence of 2 affine spaces given that one is the subset of the other one?

For the sake of completeness, I would like to give you some concepts before asking the questions: For every simplex $S=<<x^{0},x^{1},...,x^{k}>>$ in $\Bbb R^{n}$, denote by $H_s$, the ...
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### Concepts of isomorphisms of linear spaces with a norm and inner product

If I have a topological space, I say that a homeomorphic map preserves the structure of this space. Thus, in order to preserve topological properties we want to have a continuous bijection with a ...
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### The number of connected components of $SL(2, \mathbb{R})$ which keep $x^2 - y^2$ invariant

Working on yet another past comprehensive exam question. Let $S$ be the set of real $2\times 2$ matrices with determinant $1$, keeping invariant the form $x^2 - y^2$. Regard $S$ as a subset of ...
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### invertible matrices connected or not

The question asks "Is the set of all 3 by 3 real invertible matrices connected or not?" My intuitive idea is that we can establish a separation consisting of matrices with positive and negative ...
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### Set of real symmetric matrices with signature $(2,1)$ is open

Let $S$ be the space of all $3\times 3$ real symmetric matrices, let $B$ be the subset of $S$ with signature $(2,1,0)$. Show that $B$ is open in $S$ in the topology of $\mathbb{R}^6$. My thoughts: ...
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### Zero set of finitely many polynomials.

Somebody asked a question earlier regarding this proof but I'm confused about a different part. I understand everything but the line "As the zero set of finitely many polynomials, $R$ is a closed ...
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### Closure and compactness of the set of real eigenvalues ​​of a real matrix.

Let $A$ be a part of $\mathcal{M}_n(\Bbb{R})$ and $B$ the set of real eigenvalues ​​of the matrix $A$. 1) Show that if $A$ is compact then $B$ is compact as well. 2) If $A$ is closed ...
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### Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
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### Are circulant matrices open

Are the set of positive definite symmetric circulant matrices open in the set of positive definite symmetric matrices?
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### Find close points by grouping points in n-dimensional space

I know I already asked a similar question over here: Finding the closest point in a set to another point in n-dimensional space: efficiently But now my question is different. I have many points ...
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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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### If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable.

If $\overline{\operatorname{Sp}}(C)=X$ and $C$ is countable, then $X$ is separable. It seems very obvious intuitive, but how to write a good solid proof? Notice I take the closure of the span (the ...
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### Finding the closest point in a set to another point in n-dimensional space: efficiently

I'm a programmer and am working on writing an efficient algorithm that, given a point P in n-dimensional space, can find the closest point from a set of points. For ...
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### Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
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### Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
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### Topological subspace in $(S^{1})^{n}$

Studying the set of solutions of a particular linear system associated to a matroid, I notice that is it possibile to determine the topology of the quotient and identify it as a subtorus of ...
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### Elementary proof that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$ [duplicate]

Prove that $Gl_n(\mathbb R)$ and $Gl_m(R)$ are homeomorphic iff $n=m$ Since $Gl_n(\mathbb R)$ is homeomorphic to an open subset of $\mathbb R^{n^2}$, this boils down to proving that two open ...
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### Convex hull of $\exp\bigl( \mathcal{M}_n(\mathbb R)\bigr)$

What is the convex hull of $\exp\bigl( \mathcal{M}_n(\mathbb R)\bigr)$ ? My attempt : lemma For $A\in\mathcal{M}_n(\mathbb C)$ there exist $P(X)\in \mathbb{C}[X]$ such that $A=\exp{P(A)}$ ...
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### A peculiar fact about 3-dimensional complex projective space

I'm working on a result for my master's thesis, that right now involves translating a proof I don't quite follow, to something that is a bit more in line with what I already know. We define ...
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### Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
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### Let $W$ be a linear subspace of dimension $n-1$ in $\mathbb R^n$. Is $W$ closed in $\mathbb R^n$? [closed]

Let $W$ be a linear subspace of dimension $n-1$ in $\mathbb R^n$. Is $W$ closed in $\mathbb R^n$? Also, is it open?
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### Let $A$ and $B$ in $O_n(\mathbb{R})$. Show that $A$ and $B$ commute.

Let $A$ and $B$ in $O_n(\mathbb{R})$ (orthogonal matrices) such that $|||B-I_n|||<\sqrt{2}$ (subordinate norm) and $A$ commute with $BAB^{-1}$. Show that $A$ and $B$ commute. My ...
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### Some questions of vectors and dense subsets

I have a couple of quick functional analysis related questions: 1.Say we have a normed space $V$ and reflexive, separable Banach space and $K \subset V$ a closed, convex, bounded subset of $V$. ...
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### chosing between matrix theory and combinatroics

I have to take one more math course to finish my math minor , i am a computer science major and i want to know which course will benefit me more matrix theory or combinatorics and which takes more ...
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### 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 $3\times 3$ matrices and the unit circle, unit sphere, and their product ($S^1$, $S^2$, $S^1 \times S^2$, respectively). Any ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $d_{1}(x,y) = ||x-y||_{1} = \sum\limits_{i=1}^n |x_{i} - y_{i}|$.
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...