All Questions
Tagged with vector-spaces field-theory
222
questions
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Linear space and vector space correlation
I'm confused with definition of vector space and field.
According to wiki vector_space and field.
Vector space over some field is defined as set of element in $V$ and binary operations that satisfies:
...
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82
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"field" vs. "vector field" [duplicate]
Is the "field" in the "vector field" as the same "field" in algebra: as the commutative ring with the multiplicative inverse?
If yes, then the "vector field" ...
- 5,969
1
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29
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Sub division rings of dimension 2 of division rings
Suppose $A$ is a division ring and $B$ is a sub division ring such that $A$, as a left vector space over $B$, has dimension $2$.
Is it true that $B$ must be commutative ?
- 799
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Dimension of division ring over a sub division ring
Let $L$ be a division ring ("skew field") and $K$ a sub division ring. Now suppose that $L$, as a left vector space over $K$, has finite dimension $m$. Does $L$, as a right vector space over ...
- 799
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Directional derivatives in the complex plane as a vector space.
Consider the directional derivative in the direction $v\in V$ on some vector space $V$
\begin{equation}
f_v'(x)=\lim_{h\rightarrow 0}\frac{f(x+hv)-f(x)}{h}.\qquad(1)
\end{equation}
Consider $\mathbb{C}...
- 241
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What does it mean, The space $L_n (V^n;K)$ of alternating n-linear forms is of dimension one?
Reading about the fundamental theorem of alternating applications which says
Given 2 vector spaces over $K$, $(V;K)$ and $(W;K)$. If $dim\ \ V=n$ and a base of V is {$u_1...u_n$}
I saw that there is ...
- 29
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580
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When is a field "compatible" with an abelian group?
Let $F$ be a field and $G$ an abelian group. We say that $F$ and $G$ are "compatible" if and only if there exists a function $M:F\times G\to G$ such that
$M(\lambda,gh)=M(\lambda,g)M(\...
- 5,040
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81
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How to prove this is an isomorphism?
Be $\mathbb{K}$ a field, and $\alpha_0, \ldots , \alpha_n \in \mathbb{K}$ are distinct elements. Show the application
$$\mathbb{K}[x]_{\leq n} \to \mathbb{K}^{n+1}$$
$$p \mapsto (p(\alpha_0), \ldots, ...
- 3,482
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68
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Compute $d(n) = [\mathbb{F}_3[a] : \mathbb{F}_3]$ where $a\in K^\times$ has order $n$ and $K$ is a field with $81$ elements.
I know that the group of units $K^\times$ is cyclic of order $80$, so the possible values of $n$, the order of $a$, are $1,2,4,5,8,10,16,20,40,80$, the divisors of $80$. Now I am tempted to think that ...
- 173
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A group closed under scalar multiplication is a vector space
Suppose $G$ is an abelian group closed under scalar multiplication with elements in the field $F$. Is $G$ always a vector space over $F$?
I have been trying to find a counter-example, but failing. In ...
- 69
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148
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A vector space over an infinite field cannot be the union of proper subspaces: Wrong argument?
I'm trying to solve the following problem:
Suppose that $K$ is an infinite field, and $V$ is a vector space over $K$. Show that it is not possible to write $V=\bigcup_{i=1}^{n}U_i$, where $U_1,...,...
- 103
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71
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Why does the popular algorithm for for expressing the field extension F(a) as a vector space actually produce the correct answer?
When asked to describe the field extension F(a), where F is a subfield of E and a is an element of E, many people instinctively respond with the following algorithm: they generate (in E) the sequence $...
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56
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Dimension of $\mathbb{Q(\omega)}$ and minimal polynomial of $\sqrt[3]{2}$
Consider:
$$\omega = \frac{-1}{2} + \frac{\sqrt{3} i}{2}$$
and the simple extension $\mathbb{Q(\omega)}$. Find the dimension of $\mathbb{Q(\omega)}$ and the minimal polynomial of $\sqrt[3]{2}$ over $\...
- 45
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Confusion on Scalar addition in vector space
Consider this problem.
Let $V$ be the set of all vectors which are positive rational numbers on which addition of vectors $v,w$ is defined as:
$$v+w=vw$$ and the scalar multiplication is the usual ...
- 13.2k
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Why does extending scalars from $R$ to $S$ preserve isomorphism?
I am looking at a proof and there is a step I don't understand well. Sorry if this post seems disorganized, but I'm not sure how to ask this question in any other way.
Statement: Let $R$ be a ...
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