Questions tagged [vector-space-isomorphism]
This tag should be used for questions about isomorphisms between vector spaces.
658
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Computational complexity of colored vector space isomorphism over the binary field
Suppose there are two vector spaces $V$ and $W$ of the same dimension $n$ over the binary field $\mathbb{Z}_2$ together with subsets $S_V$ and $S_W$ of $V \setminus \{0\}$ and $W \setminus \{0\}$, ...
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Is this sufficient to show the two spaces are not isomorphic?
Yesterday I was studying about isomorphisms between vecto spaces, and basically the fundamental note I highlighted stated that "two vector spaces whose dimensions are different cannot be ...
- 2,748
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Is $\mathbb{C}^{2}$ isomorphic to $\mathbb{C}$ over Rational numbers [duplicate]
We can say that $\mathbb{C}^{2}$ is not isomorphic to $\mathbb{C}$ when both are considered as Vector spaces over the field of Complex numbers or Real numbers. But is $\mathbb{C}^{2}$ isomorphic to $\...
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Canonical isomorphism of $[\hom(V,W)]_{\mathbb{C}}\cong\hom(V_{\mathbb{C}},W_{\mathbb{C}})$
From proposition $8.1.3$ of "Terrence Napier, Mohan Ramachandran-An Introduction to Riemann Surfaces".
Let $V$ and $W$ be real vector spaces.
$(a)$ The set $V⊕V$, together with the standard ...
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Isomorphisms between sub division rings of degree $2$
Let $A$ and $B$ be two isomorphic division rings, both contained in the division ring $C$, and suppose that $[C : A] = [C : B] = 2$.
Under which assumptions do we know that there exists an ...
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82
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Isomorphism of a dual space
If a space $X$ is isomorphic to a dual space $Y$, then is that mean $X$ is also a dual space?
$X$ is isomorphic to $Y$ means there exist a linear map $T : X \rightarrow Y$ such that both T and $T^{-1}$...
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Show that the space of polynomials with real coefficients and $\mathbb{R}^{n+1}$ are isomorphic
From the notes from my linear algebra course:
Given a positive integer n, the vector spaces $\mathbb{P}_n(\mathbb{R})$ and $\mathbb{R}^{n+1}$ are isomorphic.
[To show this, define] $T: \mathbb{P}_n(\...
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Does $\mathbb{R}^n\times\mathbb{R}^m$ imply $\mathbb{R}^{n+m}$ mapping?
This might be a very basic question, but does the function
$$
f:\mathbb{R}^n\times\mathbb{R}^m\to \mathbb{R}
$$
necessarily imply the existence of a function
$$
g:\mathbb{R}^{n+m}\to\mathbb{R}?
$$
If ...
- 3,095
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Are irreducible elements of the tensor product of a vector space equivalent to irreducible polynomials?
I'm looking for some feedback on a construction I came across. Loosely, it entails sending a vector space isomorphically to a vector space of polynomials, 'restoring' the ring structure, and asking ...
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Algebraic meaning of the roots of a polynomial in the context of linear algebra, specifically linear transformation
A polynomial of order n $p(x) = \Sigma_{i=0}^n c_i x^i$ is isomorphic to $R^n$. So is there any algebraic meaning of the roots of a polynomial in the context of linear algebra, specifically linear ...
- 1
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Isomorphisms of the tensor product of $\mathcal{A}^N$ [duplicate]
Let $\mathcal{A}$ be an algebra and define $\mathcal{A}^N:=\mathcal{A}\oplus\dots\oplus\mathcal{A}$.
I need to show that
$\mathcal{A}^N\otimes_\mathcal{A} \mathcal{A}^N\cong M_N(\mathcal{A})$
$\...
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Is $\Bbb R^2 \subset P^2$ by isomorphism?
According to page 10 of Construction of Number Systems by Prof. Kumar, we define a map $f:\Bbb N \rightarrow \Bbb Z$ by $f(k)=[(a+k,a)]$ for some $a\in\Bbb N$. Then we show this function $f$ is one-to-...
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Relation between quaternions, Pauli vectors and regular vectors
I'm learning about rotations in 3D space. I've come across Pauli vectors and quaternions. Now as I understand, one can associate to a regular vector $\vec{v} = (v_x, v_y, v_z)$ a Pauli vector and a ...
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Condition on linear transformations to be non-singular
Let $T$ be a linear transformation over a linear vector space $L$. For $T$ to be non-singular, it has to be one-to-one and onto i.e. a bijection. Since a linear transformation is a homomorphism of the ...
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How is the arithmetic multiplication related to tensor products?
Mathematical objects are supposed to "model things", sometimes, more mathematics. In this sense, i think is "natural" to assume that tensors, being such universal objects as they ...
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proof that affine spaces are isomorphic to vector spaces?
Can someone please explain in detail the following proof that affine spaces are isomorphic to vector spaces?
To clarify the notations the book has previously defined $R_{v}$ and $v_{x,y}$ as follows
...
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If $A$ is a bounded linear operator and positive on a Hilbert space. Show that $A+\lambda I$ is an isomorphism $\forall\lambda>0$.
Let $ H $ be a Hilbert space and let $ A\in \mathcal{B}(H) $ be a positive linear operator, that is, $\langle Ax,x \rangle \geq 0$ for all $ x\in H$. Show that $A+\lambda I$ is an isomorphism, for all ...
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Dual of the annihilator isomorphic to quotient space
There are many resources online detailing the isomorphism between the annihilator $U^0:=\{f\in V':f(U)=0\}$ and the dual of the quotient space, $(V/U)':=Hom(V/U,\mathbb{R})$.
However, my lecture notes ...
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Confusion on proof of existence of ismophism between bilinear form on $L$ and dual space $\mathcal{L}(L,L^*)$
Theorem: There exists an isomorphism between the space of bilinear forms $\varphi$ on vector space $L$ and space $\mathcal{L}(L,L^*)$ of linear transformations $\mathcal{A}:L\mapsto L^*$.
I'm ...
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$(\ell^{1})^{*}$ isometrically isomorphic to $\ell^{\infty}$ as a corollary of Riesz representation Theorem.
I'm following "A Course in Functional Analysis, Conway" and as a corollary of the Riesz theorem (Example 5.9) he states what I have written in the title.
If I consider $\mathbb{N}$ with the ...
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Clifford algebra $Cl_1 \cong \mathbb{C}$
I'm reading Spin Geometry by Lawson and Michelsohn. In general Clifford algebras are defined by $Cl(V,q)=\mathcal{T}(V)/\mathfrak{I}(V)$, where $\mathcal{T}(V)$ is the tensor algebra of $V$ and $\...
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Relation between $\mathbb{Q}$ and $\mathbb{Z}^2$
Is there a way of writing $\mathbb{Q}$ in terms of $\mathbb{Z}^2$ in a simple expression?
My idea is to construct $\mathbb{Q}$ a quotient set of $\mathbb{Z}^2$, where $\mathbb{Z}$ is the set of ...
- 3,095
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Isomorphism theorems to conclude that cokernel is finite dimensional
Are the following implications correct:
let $U,V$ be two subspaces of $H$ then
$$
\frac{H}{U} = \frac{U+V}{U} \cong \frac{V}{U \cap V} \text{ and } \frac{H}{V} \cong \frac{H}{U \cap V } \Big/ { ...
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Assume $A \oplus B \cong A \oplus C$. Is $B \cong C$? [duplicate]
Let $E$ be a vector space. Let $A,B,C$ be vector subspaces of $E$. Let $\oplus$ denotes the direct sum of vector spaces. Let $\cong$ denotes the vector space isomorphism. If $B \cong C$ then $A \oplus ...
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Dimension, Orbits, G-morphisms, Function Spaces, Finite Groups
I am reading Group Theory and Physics by Sternberg. On page 64, Section 2.5 Actions on function spaces, he states that:
$$ \text{dim} \space \text{Hom}_{G}(F(M),F(N))$$ is the number of orbits of $G$ ...
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Suppose $T_1, T_2\in L(V,W)$. Prove that if $\text{null }T_1=\text{null }T_2$ then there exists invertible operator $S\in L(W)$ such that $T_1=ST_2$
The following problem appears in Ch. 3 section 3.D of Axler's Linear Algebra Done Right
Suppose $W$ is finite-dimensional and $T_1, T_2\in L(V,W)$. Prove that $\text{null }T_1=\text{null }T_2$ if and ...
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Applying the double dual isomorphism $V\cong V’’$
So, I’ve shown that a finite-dimensional vector space $V$ is naturally isomorphic to its double dual $V’’$, ie $V\cong V’’$.
My question is why is this useful? When does this come in handy? What are ...
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If $V\cong W$ and $U\subseteq V$, is $U\cong T(U)$?
In linear algebra, I’ve just shown that $V\cong V’’$, ie $V$ is isomorphic to its double dual.
I now want to show that if $U$ is a subspace of $V$ then $U$ is mapped isomorphically to $U^{00}$, ie the ...
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Can we prove that $\psi : V \rightarrow V^{**}$ is an isomorphism by constructing an inverse map?
Let $V$ be a finite dimensional vector space, and $\psi : V \rightarrow V^{**}$ be a map defined by
$$ \psi(v)(f) = f(v)$$
for all $v \in V$ and $f \in V^{**}$.
We can easily show that $\psi$ is a ...
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Proof that f : V* ⊗ W −→ HomK (V, W ) is an isomorphism
If K is a field and V,W vector spaces how to prove that there is a function f : V* ⊗ W −→ HomK (V, W ) uniquely defined by the equation f (α ⊗ w)(v) = α(v)w and f is an isomorphism (if V and W are ...
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Isomorphism Between the Sequence Spaces $c$ and $c_o$
I have been looking a little closer at the following:
There exists a linear homeomorphism $T:(c, \|$ $\left.\|_{\infty}\right)
\rightarrow\left(c_o,\|\cdot\|_{\infty}\right)$, where $c$ and
$c_{0}$ ...
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What does restricting to an isomorphism mean in this question?
Let $\mathscr{V}$ and $\mathscr{W}$ be finite-dimensional vector spaces. The nullspace of a linear transformation $L: \mathscr{V} \rightarrow \mathscr{W}$ is the set of $X \in \mathscr{V}$ such that $...
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Relationship between Noether's First Isomorphism Theorem and Dual Spaces
I have been studying about these topics. Noether's First isomorphism theorem states that if $f:V\to W$ is linear, then there exists an isomorphism between $V/\text{Ker}(f)$ and the subspace $\text{Im}(...
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For self-adjoint automorphism $T$, to what extent is $U \circ T \circ S$ equivalent to $U \circ S$?
I would like to know to what extent can composition with self-adjoint isomorphism affect composition of linear operators. More specifically, for linear operators of Hilbert spaces as $H \xrightarrow{S}...
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For a bijection $T\in\mathcal B(B_1, B_2)$ prove $B_1\setminus \text{Ker}(T)\cong \text{range}(T)$ iff $\text{range}(T)$ is closed.
Problem: Let $B_1$ and $B_2$ be two Banach spaces and take $T\in\mathcal B(B_1, B_2)$ to a bounded linear operator between the two that is also bijective. Prove that $B_1 \setminus \text{Ker}(T)\cong \...
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Help finding specific isomorphism between $\mathbb{Q}^{2g}$ and $K^2$.
I'm reading McMullen's paper "Billiards and Teichmüller Curves on Hilbert Modular Surfaces." I am stuck on understanding isomorphism (6.2) in the "Hilbert modular varieties" ...
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How to find the isomorphism between Lie algebra $L_{0}$ and $L_{1}$?
I think the 2 algebras $L_{0}$ and $L_{1}$ are isomorphic to the Lie algebra $\mathfrak{iso}(2)$ .
I struggled to find the isomorphism $\phi$ between the Lie algebra $L_{0}$
and $L_{1}$ ?
Let $L_{0}=...
- 43
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Naming isomorphism of bounded operators on Hilbert spaces coming from isomorphism of Hilbert spaces
Suppose I have an (isometric) isomorphism of separable Hilbert spaces $i:\mathcal{H}_1 \to \mathcal{H}_2$, then I can define $I:\mathcal{B}(\mathcal{H}_1) \to \mathcal{B}(\mathcal{H}_2)$ given by
$$
...
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Does $V\rightarrow W$ isomorphism map a $T:V\rightarrow V$ eigenvector to an $S:W\rightarrow W$ eigenvector?
How would go about proving the following lemma?
Lemma:
Let $V,W$ be vector spaces over $\mathbb{F}$, $\alpha:V \rightarrow W$ be an isomorphism, and $T:V \rightarrow V, S:W \rightarrow W$ be $\mathbb{...
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Is the isomorphism $V\to\mathbb K^{[\mathcal B]}$ always continuous?
While trying to generalise the well-known result "continuous partials imply differentiability" to the case of infinite-dimensional spaces, the following question popped up in my work:
Let $...
- 3,163
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How to show that a Vector space is isomorphic with the range of a vector space
I have the following question.
Let $V$ and $W$ be vector spaces, let T: $V \to W$ be a linear transformation, and let $B = \{v_1, v_2, ..., v_n\}$ be a basis of $V$. Prove that if $T$ is injective, ...
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Proving that range(T) $\cap$ ker(T)={0} for a linear transformation T: V $\rightarrow$ V
My question is related to this other question:V is a n-dimensional vector space and $T : V \rightarrow V$ is a LT s.t $rank(T) = rank(T^2)$. Prove that $range(T)\cap ker(T) = \{\mathbb{0}\}$. I solved ...
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Looking for a general theorem in Linear Algebra about the isomorphism between linear functions and matrices
I'd be thankful for a reference to the most general form of this theorem. If I'm not mistaken, it is true that
$$
\mathcal{L}(U_n, V_m) \cong M_{m\times n}(\mathbb{F}),
$$
where $U_n$ is a $n$-...
- 2,019
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Isomorphism from $k\times n$ matrices to $\operatorname{Hom}(K^{n\times 1}, K^{m\times 1})$ [duplicate]
Let $K$ be a field and $n,m\in \mathbb{N}.$ Show that the linear map $$K^{m\times n} \rightarrow \text{Hom}(K^{n\times1},K^{m\times1}); \ A\mapsto A'.$$
The linear map $A'$ is defined as $A':K^{n×1}→K^...
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1
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51
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Is being isomorphic to dual hereditary?
Let $V$ be a vector space, and assume that $V$ is isomorphic to its dual, i.e., $V \simeq V^*$. Is every linear subspace $U$ of $V$ also isomorphic to its dual, i.e., $U \simeq U^*$?
This is certainly ...
- 4,897
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Provide a injection from tangent to ambient space
I trying to translate some concepts to formalism of differential geometry.
Context
I started with a function $f:\mathbb R^{1+N}\rightarrow\mathbb R^m$ and I have defined the set
$$
Q \equiv \{(t, x) \...
1
vote
1
answer
67
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Explain Proof on Banach-Mazur distance
There are two (Why?) in the proof, i think they are trivial but, I don't see why they are true.
If anyone can show me some proof of these it would be very helpful.
Definition: Let $X,Y$ two normed ...
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Show that this mapping is an isomorphism between the vector space $\mathcal{M}_{\mathbb{K}}(n,p)$ and $L(\mathbb{K}^p,\mathbb{K}^n)$
I am considering the mapping $ u : \mathcal{M}_{\mathbb{K}}(n,p)\to L(\mathbb{K}^p,\mathbb{K}^n) : M\mapsto l_{M} $ and I would like to show this is a vector space isomorphism.
I should mention that $...
- 1,262
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Show that $\mathbb{R}^d$ is isomorphic to $\mathbb{R}^{k}\times\mathbb{R}^{d-k}$ for $d>k>0$ integers.
I would like to show this, here is my starting point
Consider the mapping $f : \mathbb{R}^d\to\mathbb{R}^{k}\times\mathbb{R}^{d-k}$
defined as $f(v)=((v_1,...,v_k),(v_{k+1},...,v_d))$
Then we have $f(\...
- 1,262
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1
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Intuition on Banach-Mazur distance
Suppose two n-dimensional normed vector spaces $X,Y$ are isomorphic, and we define the Banach-Mazur distance between $X,Y$ as $$ d(X,Y)=\inf \{ \|T\|\|T^{-1}\|:T\in GL(X,Y) \} ,$$ where $GL(X,Y)$ is ...
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