Questions tagged [vector-space-isomorphism]
This tag should be used for questions about isomorphisms between vector spaces.
683
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Canonical proof of isomorphisms like $\operatorname{Hom}(V \otimes V, V \otimes V) \cong V^* \otimes V^* \otimes V \otimes V$
$\def\Hom{\operatorname{Hom}}$
$\def\F{\mathbb{F}}$
$\def\qty#1{\left(#1\right)}$
$\def\tu{\tilde{u}}$
$\def\tv{\tilde{v}}$
$\def\vx{\vec{x}}$
$\def\vy{\vec{y}}$
To show that $\Hom(V,V) \cong V^* \...
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If two infinite dimensional vector spaces are isomorphic, does an inner product isomorphism exist between them? [closed]
The following proposition is true:
If $V$ and $W$ are finite dimensional vector spaces over a field $\mathbb{F}$, the following is equivalent.
$\quad$ (1) $V$ and $W$ are isomorphic as inner product ...
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Does "shift of terms" in a Faber–Schauder series expansion of f ∈ C[0,1] produce an element of C[0,1]?
Question
Let $\{e_n : n=0, 1, \dots \}$ be a Faber–Schauder basis of $C[0,1]$ (see: Example 4.1.11 in [1]). Is the following function well-defined:
$$
\begin{align}
T \colon C[0,1] \times \mathbb{R} &...
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Dual space isomorphism non-canonical choice example
In a lot of resources that I have read it is mentioned that the isomorphism between $V$ and $V^*$ is non-canonical, but I was never sure that I properly understood precisely what this means. I haven't ...
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$\ell^{p}(I, Z \oplus Y) \approx \ell^{p}(I, Z) \oplus \ell^{p}(I,Y)$
In "The decomposition technique" by Pełczynski one uses that
$\ell^{p}(I, Z \oplus Y) \approx \ell^{p}(I, Z) \oplus \ell^{p}(I,Y)$
where $\ell^p(I,X)$ is the infinite direct sum formed by ...
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$X \approx \ell^p(I,X)$ $\Rightarrow$ $X \approx X \oplus X$ (used in The Pełczynski decomposition technique)
In "A. Pełczynski, Projections in certain Banach spaces. Stud. Math. ´ 19, 209–228" the following relation is used as a fact (for The Pełczynski decomposition technique):
$X \approx \ell^p(I,...
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Isomorphism between $\ell^p(\mathbb{N} \times \mathbb{N}, \mathbb{R})$ and $\ell^p(\mathbb{N}, \mathbb{R})$
to apply the the Pełczynski decomposition technique I want to show that the infinte direct sum of $\ell^p$ is ismoetric isomorph to $\ell^p$. With infinte direct sum I mean:
$\ell^p(X) = \{(x_n)_{n=1}^...
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External direct sum $U_1\oplus U_2$ isomorphic to $U_1+U_2$ does not necessarily imply that $U_1\cap U_2 = \{0\}$ [closed]
Let $V$ be a vector space (not necessarily finite-dimensional) and let $U_1,U_2\subset V$ be subspaces. If $U_1\cap U_2=\{0\}$, then the surjective linear map $\phi\colon U_1\oplus U_2 \to U_1+U_2$ ...
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How to construct a canonical isomorphism between vector spaces?
Is there any general idea one must know in order to construct a canonical isomorphism between two vector spaces? I do understand that a canonical isomorphism is an isomorphism that is independent of ...
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1
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Proving that outputs of isomorphism V --> W form a basis in W
Let $T$ be an isomorphism from vector space $V$ to $W$. Suppose $S=\left\{\vec{v}_1, \cdots, \vec{v}_n\right\}$ is a basis for $V$. Show that $\left\{T\left(\vec{v}_1\right), \cdots, T\left(\vec{v}_n\...
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Linear version of the submersion theorem
I'm trying to prove the constant rank theorem for smooth functions in euclidean spaces and I've stumbled into a problem:
Given a direct sum decomposition $\mathbb{R}^{n+m} = E \oplus F$ such that $\...
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A subtle point about vector space isomorphisms
So I was studying tensor products from the book "An Introduction to Tensors and Group Theory for Physicists". After proving the fact that $ \{ e_i \otimes f_j\}_{i \in \mathcal{I}, \, j\in \...
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For all $n \in \mathbb{N}^*$, prove that the set $\left(e^{ kx }\right)_{k \in [|1,n|]}$ is linearly independent.
$(\mathbb{R}^{\mathbb{R}},+, \cdot)$ is an $\mathbb{R}$ Vector space.
For all $n \in \mathbb{N}^*$, prove that the set $\left(e^{ kx }\right)_{k \in [|1,n|]}$ is linearly independent.
I have already ...
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Let $Y$ and $Z$ be subspaces of the finite dim vector spaces $V$ and $W$, respectively. Let $R=\{α\in L(V,W):α(Y)\subset Z\}$. What is $\dim R$? [closed]
How would I go about proving this? I tried by considering maps from $Y$ to $Z$ and the space of all such maps would have $\dim Y\times\dim Z$ but I have no idea how to extend it to the entire vector ...
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$C[0,1]/C_0$ is isomorphic to $\Bbb R$
Let
$$ C_0= \{f \in C[0,1] \mid f(1/2)=0 \} $$
then $ C_0$ is a subspace of $C[0,1]$ and $C[0,1]/C_0$ is isomorphic to $\Bbb R$.
I have proved that $ C_0 $ is a vector subspace.
I tried to prove that $...
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Understanding subspace-restricted isomorphisms and global isomorphisms
(It may be noted that the author was not aware of Endomorphisms & Automorphisms at the time of writing this question)
I'm trying to better understand the concept of a linear transformation acting ...
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Isomorphism between two general linear group. [closed]
If $V$ is a vector space over the field $F$, the general linear group of $V$, written $GL(V)$, is the set of all bijective linear transformations $V\to V$, together with functional composition as ...
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Is the tensor product of two vector space on $R$ isomorphic to $R^{d^2}$?
Let $V$,$W$ be two fine-dimensional vector spaces over the field of real numbers $\mathbb{R}$. Assume the dimension of both spaces is d.
Is there a unique isomorphism between $V \otimes W$ and $\...
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isomorphism between two random vector subspaces
I have this problem which is the following :
Let K be a field, $E$ a K-vector space (finite or infinite dimension) and $F$,$G$ two vector subspaces of E. Now, consider S to be a common supplementary ...
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Isometric Isomorphism From $\mathbb{C}^n \oplus M([a,b])$ Onto $(C^n[a,b])^*$
I have the following norm in $C^n[a,b]$:
$$\| f \| = \sum_{k=0}^{n-1}|f^{(k)}(a)| + \sup_{[a,b]} |f^{(n)}|$$
I want to show that:
$$(\lambda_0,...,\lambda_{n-1},\mu)\mapsto \left( f\mapsto\sum_{k=0}^{...
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Is an isomorphism between a finite-dimensional vector space and any vector space enough to justify finite-dimensionality?
Notation.
$\mathcal{L}(V,W)$ is the set of linear maps from a vector space $V$ to a vector space $W$.
$\mathbf{F}^{m,n}$ is the set of $m$-by-$n$ matrices whose entries are either real or complex ...
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How does $W \subset W^{00}$ ? Hoffman and Kunze theorem 3.18
Hoffman and Kunze theorem 3.18 . If $S$ is any subset of a finite-dimensional vector space $V$,
then $(S^0)^0$ is the subspace spanned by $S$.
$S^0$ is the annihilator of $S$ and $S^{00}$ is ...
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Isomorphism from a set of $n\times n$ matrices with trace $x$
I am wondering how I would find an isomorphism from a set of matrices of dimension $n\times n$ with a specific value for the trace. For simplicity, let us say the trace must be equal to $1$, although ...
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Axler: Prove $\mathcal{L}(V_1\times\ldots\times V_m,W)$ and $\mathcal{L}(V_1,W)\times\ldots\times \mathcal{L}(V_m,W)$ are isomorphic vector spaces. [duplicate]
Suppose $V_1,...,V_m$ are vector spaces. Prove that $\mathcal{L}(V_1\times\ldots\times V_m,W)$ and $\mathcal{L}(V_1,W)\times\ldots\times \mathcal{L}(V_m,W)$ are isomorphic vector spaces.
There are ...
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Proof that solutions can be "translated" between two isomorphic spaces
When we face an analytic problem, It is generally possible to represent it on the Cartesian plane, solve it, and than convert the solution back to the related algebric solution.
I guess that we can do ...
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If 2 vector space are isometric then they are isomorphic
We call a map between 2 normed vector spaces is isometric isomorphism if it is an isometry and an linear isomorphism. Then, these 2 vector spaces are called isometrically isomorphic.
Suppose $X,\,Y$ ...
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isometric isomorphism implies diffeomorphism
Suppose $E$ be a $\mathbb R-$vector space that is isometrically isomorphic to $\mathbb R^n$ via the map $f:\, E\to\mathbb R^n$, which means $f$ is bijective and
\begin{align}
d_E(x,y)&=d_{\mathbb ...
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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 ...
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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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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 ...
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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 ...
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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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3
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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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2
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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 ...
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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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