All Questions
Tagged with vector-spaces abstract-algebra
1,164
questions
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Questions about how to show $d_1+\cdots +d_n-n+1 \leq {\text{dim}}_k k[x_1,\ldots,x_n]/\mathfrak{a}\leq d_1d_2\cdots d_n\quad $
The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\...
- 3,683
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1
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71
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Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.
The following are from Froberg's Introduction to Grobner bases, and Hungerford's undergraduate Abstract Algebra text.
Background
Theorem 1: $k[x_1,\ldots,x_{n-1},x_n]\backsimeq (k[x_1,\ldots,x_{n-1}])...
- 3,683
-1
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1
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Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
- 87
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Confusion over tensor definition of exterior power of a vector space and exterior algebra
I am new to and currently learning about Tensor Algebra and Exterior Algebra. I am confused about the definition of the exterior power of a vector space $V$, $\textstyle \bigwedge^k (V)$, and the ...
- 21
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1
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63
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Representing the finite field as $\{i*g+j\}$ where $g$ is a generator
This question arose from my thoughts on why the size of a finite field is always a prime power like $p^n$.
First, $\Bbb Z/p\Bbb Z$ is a field, and $\Bbb Z/p\Bbb Z -\{0\}$ is a cyclic group under the ...
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1
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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
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74
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Complexification of a vector space $V$
The tensor product $V \otimes \mathbb{C}$ is formed by taking the real vector space $V$ (where $\dim V=n$) and extending its scalars from $ \mathbb{R} $ to $ \mathbb{C} $. Elements in $ V \otimes \...
- 874
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Question about Artin's Algebra Example 3.3.4 on vector space
Example 3.3.4 Let $F$ be the prime field $\mathbb{F_p}$. The space $F^2$ contains $p^2$ vectors, $p^2-1$ of which are nonzero. Because there are $p-1$ nonzero scalars, the subspace $W=\{cw\}$ spanned ...
- 1
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Why must a reducible KG-module of dimension 2 have a KG-submodule of dimension 1?
I'm trying to see this with the Quaternion group $Q_8 = \langle a,b \ | \ a^2=b^2,\ bab^{-1} = a^{-1} \rangle $, and the representation $\rho: Q_8 \rightarrow GL_2 (\mathbb{C}) $ defined by $\rho(a)=...
- 1
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1
answer
42
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Discriminant of a matrix with respect to change of basis
According to "Commutative Ring Theory" by Matsumura:" If $A$ is a finite $k$-algebra, the trace of an element $\alpha$ of $A$ denoted by $tr_{A/k}(\alpha)$ is the trace of the $k$-...
- 420
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1
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Counting maximal subgroups of $\mathbb{Z}_m^n$
Let
$$
\mathbb{Z}_m=\mathbb{Z} / m\mathbb{Z}
$$
How many maximal subgroups does
$$
\mathbb{Z}_m^n=\underbrace{\mathbb{Z}_m \times \mathbb{Z}_m \times \cdots \times \mathbb{Z}_m}_n
$$
have? (m need ...
- 163
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49
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Set of polynomials and dimension stability under addition
I consider the subset $\mathcal{U}\subset\mathbb{R}_{2}[x_1,x_2]\times \mathbb{R}_{2}[x_1,x_2]$ defined by
$$
\mathcal{U} = \{(x_1,x_2)\in[0,1]^2\mapsto (ax_1,b(1-x_1)x_2 : (a,b)\in\mathbb{R}^2\}
$$
...
- 1,381
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relation between $E[x]$-module and $F[x]$-module when $E$ is a subfield of $F$
Let $F$ be a field and $E$ be a subfield $F$. Let $X$ be a $n\times n$ matrix with entries in the field $E$. We can give the $n$-dimensional vector space $E^n$ over $E$, a $E[x]$-module structure by ...
- 87
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4
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Which vector space axiom(s) is (are) going to fail if we take $\mathbb{C}$ as our set of vectors and $\mathbb{Z}$ as our set of scalars?
Here is the standard definition of vector space:
Let $F$ be a field, and let $X$ be a non-empty set such that
(A0) for each pair $x, y$ of elements of $X$, there exists a unique element $x + y$ in $...
- 26.9k
4
votes
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Representation of $V$ as $\mathbb{C}^{2}$
Let $V$ be a finite-dimensional complex inner product space and suppose that there is an operator (a matrix) $A$ on $V$ that satisfies the following anti-commutation relations:
$$AA + AA = 0$$
$$A^{*}...
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