Fix a field $K$. Given a vector space $V$ with a basis $B$, a vector space $W$ with a basis $C$ and a linear map $f: V \to W$, let $\{f_{c, b}\}_{c, b}$ be the representing matrix of $f$, meaning that $$ f\left(\sum\limits_{b \in B} \lambda_b b \right) = \sum\limits_{c \in C} \left( \sum\limits_{...

In chapter 19 of Goldblatt's Lectures on the Hyperreals he discusses Hyperfinite Approximation. This phenomenon suggests a new methodology for analysing infinite structures by "lifting" a corresponding analysis that is known for finite ones. In section 19.8 he discusses Hyperfinite Dimensional ...

In chapter 19 of Goldblatt's Lectures on the Hyperreals he discusses Hyperfinite Approximation. This phenomenon suggests a new methodology for analysing infinite structures by "lifting" a corresponding analysis that is known for finite ones. In section 19.8 he discusses Hyperfinite Dimensional ...

In chapter 19 of Goldblatt's Lectures on the Hyperreals he discusses Hyperfinite Approximation. This phenomenon suggests a new methodology for analysing infinite structures by "lifting" a corresponding analysis that is known for finite ones. In section 19.8 he discusses Hyperfinite Dimensional ...

I want to prove the following question: Let $B$ be a basis of a matroid $M.$ If $e \in B,$ denote $C_{M^*}(e, E(M) - B)$ by $C^*(e,B)$ and call it the fundamental cocircuit of $e$ with respect to $B.$\ $(a)$ Show that $C^*(e,B)$ is the unique cocircuit that is disjoint from $B - e.$\ $(b)$ If $f ...

I want to prove the following question: Let $B$ be a basis of a matroid $M.$ If $e \in B,$ denote $C_{M^*}(e, E(M) - B)$ by $C^*(e,B)$ and call it the fundamental cocircuit of $e$ with respect to $B.$\ $(a)$ Show that $C^*(e,B)$ is the unique cocircuit that is disjoint from $B - e.$\ $(b)$ If $f ...

Here is the question I am trying to solve (Matroid Theory, second edition by Martin Oxley, Chapter 1, section 2): Prove that if $C$ is a circuit of a matroid $M$ and $e \in C,$ then $M$ has a basis $B$ such that $C = C(e, B).$ My thoughts: My idea for solving this question is that, given a circui...

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?

Here is the question I am trying to solve (Matroid Theory, second edition by Martin Oxley, Chapter 1, section 2): Prove that if $C$ is a circuit of a matroid $M$ and $e \in C,$ then $M$ has a basis $B$ such that $C = C(e, B).$ My thoughts: My idea for solving this question is that, given a circui...

Let $(X,d)$ be a metric space. For all points $x,y \in X$ we define the metric segment between them as the following set: $$\left [ x,y \right ] = \left \{ z \in X : d(x,z)+d(z,y)=d(x,y)\right \}$$ We then say that a set $S\subseteq X$ is convex if for all $x,y \in S$ it holds true that $\left [...

Let $X = \mathbb{R}^2$ with $(x, y)\in X$ for $y\neq 0$ isolated and $(x, 0)$ having neighbourhood basis of the form $$U_n(x) = \{(x, y) : y\in (-1/n, 1/n)\}\cup \{(x+y+1, y) : 0 < y < 1/n\}\cup \{(x+\sqrt{2}+y, -y) : 0 < y < 1/n\}.$$ The space $X$ is called Mysior plane, and it's an example of a...

I have these subsets of $R^n$. $A=\{B(x,r): x \in{R^n}, r \in Q^+\}$ $B=\{B(x,r): x \in{Q^n}, r \in Q^+\}$ $C=\{B(x,r): x \in{R^n}, r > 1 \}$ how can i prove are basis for the euclidean topology?

I have these subsets of $R^n$. $A=\{B(x,r): x \in{R^n}, r \in Q^+\}$ $B=\{B(x,r): x \in{Q^n}, r \in Q^+\}$ $C=\{B(x,r): x \in{R^n}, r > 1 \}$ how can i prove are basis for the euclidean topology?

Let $T$ be the linear operator on $F^2$ which is represented in the standard ordered basis by the matrix $$\begin{bmatrix}0&0\\1&0\end{bmatrix}.$$ Let $\alpha_1=(0,1)$. Show that $F^2\neq Z(\alpha_1;T)$, and that there is no non-zero vector $\alpha_2$ in $F^2$ with $Z(\alpha_2;T)$ disjoint from ...

Let $\mathfrak {g}$ be a semisimple Lie algebra with a non-degenerate invariant bilinear form $B$ (e.g. Killing form). Then what is meant by saying that $C$ is a Casimir tensor with respect to the bilinear form $B\ $? The wikipedia article on Casimir element says the following $:$ Given a basis $...

A subset $B$ is called an (asymptotic) additive basis of order $2$ if every sufficiently large natural number $n$ can be written as the sum of at most $2$ elements of $B.$ How small/sparse can such sets be asymptotically? So for example, let $b_n$ be the $n-th$ element of $B$, when $B$ is writt...

Let $k$ be an arbitrary field, and let $V$ be an arbitrary $k$-vector space, possibly infinite-dimensional. Let $g\in\operatorname{End}_k(V)$. Then: $g$ is diagonalizable if $V$ has a basis of eigenvectors for $g$; $g$ is semisimple if $g$ is diagonalizable over $\overline{k}$; $g$ is nilpoten...

Let $p:X \to Y$ be a perfect map. Then: $(4)$ If $X$ is $2^\circ$ countable, so also is $Y$. Dugundji’s proof: Let $\{U_i\}$ be a countable basis for $X$, and let $\{V_i\}$ be the family of all finite unions of $U_i$; by chapter 2, 8.8, the family $\{V_i\}$ is countable, and we show that the o...

Let $G$ be a discrete amenable group (maybe also finitely generated, if required to assume) acting on a compact, metrisable, finite dimensional topological space $X$. Let $\{U_\alpha\}_{\alpha\in I}$ be a finite open cover of $X$. Is it true that if the action is free and / or minimal, then the r...

We have a vector space $V$ with basis $\underline{\boldsymbol{v}} = (\boldsymbol{v}_{1}, \boldsymbol{v}_{2}, \boldsymbol{v}_{3}, \boldsymbol{v}_{4}, \boldsymbol{v}_{5})$, and for $1\leq i \leq 5$, we have $V_{i} = \langle \boldsymbol{v}_{1}, \ldots, \boldsymbol{v}_{i} \rangle_{\mathbb{k}}$ so tha...

The basis was recently created. But it's a horrible tag. There are different notions of basis in mathematics, and they are not entirely the same at all. Hamel basis Hilbert basis. Schauder basis. Topological basis. The tag is used as a free for all. And if it continues to exists, it will be u...

Let $(X,O)$ be topological space which is defined by local basis(fundamental neighborhood system)$B_x$ at $x∈X$. Then, why $∪_{x∈X}B_x$ is basis of $X$, in other words, every open set of $(X,O)$ is written by union of $∪_{x∈X}B_x$? I think this holds in general topological space, but I have neve...

Let $(X,O)$ be topological space which is defined by local basis(fundamental neighborhood system)$B_x$ at $x∈X$. Let $(X,O')$ be another topological space, which is defined by local basis $B'_x$ at $x∈X$. If for all $x∈X, B_x'⊂B_x$ , then two topology coincidences ? For example, open $ε$ ball an...

find the basis of $C^3 = \{(z1,z2,z3) | z1,z2,z3 \in C\}$

Let $(X,O)$ be topological space which is defined by local basis(fundamental neighborhood system)$B_x$ at $x∈X$. Then, why $∪_{x∈X}B_x$ is basis of $X$, in other words, every open set of $(X,O)$ is written by union of $∪_{x∈X}B_x$? I think this holds in general topological space, but I have neve...

Let $E$ be a set. Suppose that for each $x$ there is a non-empty collection $\mathcal{B}_x$ of subsets of $E$ such that (B1) $x\in B, \forall B\in\mathcal{B}_x$ (B2) if $B_1\in\mathcal{B}_x$ and $B_2\in\mathcal{B}_x$, then $\exists B_3\in\mathcal{B}_x$ such that $B_3\subseteq B_1\cap B_2$ (B3) i...

My uni professor has taught us the following: If the likelihood formed on the basis of a random sample from a distribution belongs to the regular exponential family, then the likelihood equation for finding the ML estimate of the parameter vector $\boldsymbol{\theta}$ is given by [equation 1]$$\m...