# Tagged Questions

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### Infinite dimensional Vector Spaces and Bases of Quotients

Given a basis $\mathcal{B}$ of an infinite dimensional vector space $V$, will a quotient of $V$ by a subspace $I$ necessarily have a basis? Is there a way of obtaining it using $\mathcal{B}$? My ...
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### Finding a suitable basis for a vector space

Let $k$ be a division ring and $V$ be a left $k$-vector space of infinite dimension. Let $E=\mathrm{End}(V)$, defined as a ring of right operators on $V$. My questions are: (1) Why $V$ is a simple ...
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### Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
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### Inclusion induce an isomorphism

I have a small question If $E,F$ are two sub-modules of a module $G$, how to prove that the inclusion $E \rightarrow E+F$ induces an isomorphism $(E/E \cap F) \simeq (E + F) / F$ Please Thank you ...
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### Is there vector space $\mathbb R$ over $\mathbb R$ with dimension $m\neq1$?

I am searching a new scalar product that with this scalar product the vector space $\mathbb R$over $\mathbb R$ with ordinary vector sum has dimension $m$ , which $m$ is any arbitrary natural ...
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### Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
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### Linear dual of vector fields

Suppose that $M$ is a smooth manifold and $\mathfrak{X}(M)$ is the set of smooth vector fields on $M$. There are basically two different linear structures on $\mathfrak{X}(M)$: 1.) $\mathfrak{X}(M)$ ...
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### Why is the Ideal Norm Multiplicative?

I had asked this question before and got a partial answer. Let $A$ be a Dedekind domain with quotient field $K$, $L$ a finite separable extension of $K$ of degree $n$, and $B$ the integral closure of ...
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### R-modules over a vector space

Let $V = \mathbb{C}^3$, $A$ be the matrix with column vectors $e3, e1, e2$ (where $e1, e2, e3$ are the standard basis vectors for $\mathbb{R}^3$). $R = \mathbb{F}[X]$. Let $V_a = V$ be the R module ...
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### Spinor representation and Clifford modules

Let $V$ be an even-dimensional real inner product space. We denote the Clifford algebra of $V$ by $C(V)$ and the spinor representation by $S$. For a finite-dimensional $\mathbb Z_2$-graded complex ...
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### There's a bijective correspondence between complements of a submodule $N \leqslant M$ and extensions of $\sigma : N \approx M_1$.

Let $N$ be a submodule of $R$-module $M$. let $\sigma : N \approx M_1$ be an $R$-isomorphism. Then the correspondence $H \to \bar{\sigma}$, where $\ker(\bar{\sigma}) = H$ is a bijection from the ...
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### A basis of a module is a minimal spanning set.

This is from Roman's Advanced Linear Algebra. Theorem 4.4 Let $\mathcal{B}$ be a basis for an $R$-module $M$. Then $(1) \ \mathcal{B}$ is a minimal spanning set. $(2) \ \mathcal{B}$ is a ...
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### Obtain a basis of invertible matrices for $M_n(D)$, where $D$ is an integral domain

Let $D$ be an integral domain. Prove that $M_n(D)$ has a basis consisting of $n^2$ invertible matrices. Consider $M_n(D)$ as a $D$-module. Define invertible elements $V_{nn}=I_n$, \$V_{n-1,n-1} = ...