All Questions
Tagged with vector-spaces modules
246
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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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27
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Revisiting proof that all bases of a free module $M$ over a commutative unitary ring are equipotent
I am following closely the book of T.S. Blyth, Module Theory. There is a theorem which says that every free $R$-module $M$ where $R$ is a commutative unitary ring, has equipotant base.
The process is ...
- 876
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32
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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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Is $\mathbb{Z}^{\mathbb{N}} \otimes_\mathbb{Z} \mathbb{Q} \cong \mathbb{Q}^{\mathbb{N}}$, non-canonically?
In general, tensor products don't commute with direct products. As such, I understand that the natural map $(\prod\mathbb{Z}) \otimes_{\mathbb{Z}} \mathbb{Q} \to \prod (\mathbb{Z}\otimes_{\mathbb{Z}} \...
- 943
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Modules and Diagonizability using Primary Decomposition
Given a field $K$, a vector space $V$ over $K$ and a linear map $T:V \longrightarrow V$ of finite dimension with minimal polynomial $\mu(x)\in K[X]$, prove using the primary decomposition of V (...
- 175
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81
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Totally isotropic subspace for bilinear pairing over ring
Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb{F}$-vector space $V$ and $W$ a totally isotropic subspace. ...
- 743
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Theorem 4, Section 4.2 of Hungerford’s Algebra
Every vector space $V$ over a division ring $D$ has a basis and is therefore a free $D$-module. More generally every linearly independent subset of $V$ is contained in a basis of $V$.
Sketch of proof: ...
- 4,249
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44
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what are the differences between Jordan Canonical forms and rational canonical forms?
I am studying JCF and RCF from Dummit & Foote. I can say what are all the differences between them, either in computations or in definitions and constituents. Could someone clarify this to me ...
- 2,087
4
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76
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sum of projections equivalence over vector space
Let $V$ be a vector space, let $p, q$ projections over $V$. I am trying to prove the statement
which says that if $p+q$ is a projection, then $p\circ q=q\circ p=0$.
I could only show that $p\circ q + ...
- 876
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78
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Showing that the rank of $M$ is exactly $1.$
Here is the question I am trying to solve:
Let $R = \mathbb Z[x]$ and let $M = (2,x)$ be the ideal generated by $2$ and $x,$ considered as a submodule of $R.$ Show that $\{2,x\}$ is not a basis of $M.$...
user965463
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2
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137
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If $\varphi|_W$ and $\bar{\phi}$ are nonsingular prove that $\varphi$ is nonsingular.
Here is the question I am trying to answer the second and the third part of it:
If $W$ is a subspace of the vector space $V$ stable under the linear transformation $\varphi$(i.e., $\varphi(W) \...
- 3,129
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83
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Why is $ V \longrightarrow V: v \mapsto \pi(g)(v)$ an endomorphism of $G$-modules?
I found this question here finite dimensional irreducible unitary representations
But I having trouble with the answer given there
Let $(\pi,V)$ be a unitary finite dimensional unitary representation ...
- 3,162
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1
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86
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Generalized Notion of Krylov Subspaces
Let $\mathcal{X}$ be a vector space over a field $\mathbb{K}$ and let $x_0 \in \mathcal{X} \setminus \{0_{\mathcal{X}}\}$ (here, $0_{\mathcal{X}}$ denotes the zero vector in $\mathcal{X}$). We denote ...
- 510
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32
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The name for a type of map between vector spaces
Is there a name for a map $f:V \to W$ between two $\mathbb{K}$-vector spaces that is not linear map but which still staisfies
$$
f(\lambda v) = \lambda f(v), ~~~~~ \textrm{ for all } \lambda \in \...
2
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1
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68
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$\mathbb{Q},\mathbb{R}$ and $\mathbb{C}$-vector space isomorphism from $\mathbb{Z}$-module isomorphism.
Let $A,B$ be $\mathbb{C}$-vector space. We can view them as a $\mathbb{Z}$-module. Suppose that there is a $\mathbb{Z}$-module isomorphism $\phi$ between $A$ and $B$. Then can we have a natural $\...
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