Questions tagged [dual-maps]
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54 questions
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Relation between the elementary Rouché-Fröbenius theorem and the more abstract Fröbenius theorem with exact sequences.
In highschool I was taught that the "Rouché-Fröbenius theorem" was that given a homogenous linear system of equations, the solution space has dimension $n-r$ where $n$ is the dimension of ...
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21
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What’s the Hopf dual of quantum supergroup Uq(gl(m|n))
My problem is if $GL_q(m|n)$ and $U_q(gl(m|n))$ are Hopf dual to each other, and if they are dual, if there are some references.
I ask this question because I saw an example in Kassel’s book: $GTM155$ ...
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Second existence theorem for weak solution of uniformly elliptic PDE in Evans
I have some questions on the proof of second existence theorem (theorem 4 in section 6.2.3 in Evans' pde book, one could have a copy of this book here). The following is abridged from step 4:
we ...
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Prove $T' = 0$ if and only if $T=0$
Theorem. Given that $W$ is finite-dimensional and $T \in \mathcal{L}(V, W)$. then $T'=0$ if and only if $T=0$.
I’m having trouble proving $T=0$ if $T' = 0$. I see a counter example to this. From ...
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Dual map-injectivity implies surjection
Let $k_n=\mathbb{Q}_p(\zeta_{p^n})$ and $T$ be the $p$-adic Tate module of an elliptic curve $E$. Assume $E$ has good supersingular reduction at $p$. Let $V=T\otimes \mathbb{Q}_p$. Now, by Local Tate ...
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163
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$\text{span}(\phi_1, \cdots \phi_m) = \{\bigcap_{i=1}^m \text{null}(\phi_i)\}^0$
Suppose $V$ is finite-dimensional and $\phi_1, \cdots, \phi_m \in V'$. Prove that the following three sets are equal to each other.
(a) $A = \text{span}(\phi_1, \cdots, \phi_m)$
(b) $B = \left(\...
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How do we know $T'(\psi_j)=\psi_j\circ T$, where $T:V\to W$, $T': W'\to W'$ is dual map of $T$, and $\psi_j\in W'$?
Sometimes I have trouble with the notation in the book "Linear Algebra Done Right" by Axler.
Currently, my issue is with the following theorem (Theorem 3.114 in the 3rd edition book)
Let $...
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0
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What’s the intuition for these annihilator results in linear algebra?
I’m studying dual spaces in linear algebra.
I have proved the following two results. Note: I say $U^0$ for the annihilation of $U$.
For subspaces $U,W$ of a vector space $V$,
$(U+W)^0=U^0\cap W^0$
And
...
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Adjoint of the Dual Map
Given a linear map $f:X\longrightarrow Y$ between finite dimensional Hilbert spaces, we can define the dual map $f^*:Y^*\longrightarrow X^*$ by $f^*(\phi) = \phi\circ f$. In a paper on quantum ...
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If two elements are different there is a functional under where the image is different
I have the following exercise in functional analysis:
Let E be a normed space and $x,y$ different vectors. Prove or disprove finding a counterexample that it exist a function $f \in E^*$ such that $f(...
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548
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Can someone explain why we define adjoint?
Recently I've been reviewing linear algebra. The definition of adjoint of a linear map on an inner product space seems not really natrual. Looks like people use this to define normal and self-adjoint ...
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Why is $\mathrm{range}(T^*)=(\mathrm{null}T)^0$?
I'm reading Axler's Linear Algebra Done Right Third Edition and I'm struggling to figure out why exactly $\mathrm{range}(T^*)=(\mathrm{null}(T))^0$ by theorem 3.109 where $T\in L(V,W)$ and $V,W$ are ...
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Characterization of duality map
I'm doing Ex 1.1.3 in Brezis's book of Functional Analysis. Could you have a check on my attempt?
Let $E$ be a n.v.s. and $E'$ its dual. The duality map $F$ is defined for every $x \in E$ by
$$
F(x) =...
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The property of duality map in case the dual space is strictly convex
Let $(E, | \cdot |)$ be a Banach space and $(E', \| \cdot \|)$ its dual. Assume that $E'$ is strictly convex. Then for each $x\in E$, there is a unique $f_x \in E'$ such that $\|f_x\|=|x|$ and $\...
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Is the transpose of $T$ also the dual map of $T$?
Suppose I have $T: V \to W$. Then $T$ defines a map $T': W' \to V'$ that some sources (e.g. Axler) call the dual map of $T$. And $T$ defines a map $T^t: W' \to V'$ that some sources (e.g. Hoffman and ...
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Defining De Morgan dual using $\neg: \mathbf{A}^{\text{op}} \to \mathbf{A}$
Preamble. Suppose $\textbf{A} = \langle A,\bot,\top,\vee,\wedge,\neg\rangle$ is a Boolean algebra. The "opposite" Boolean algebra $\textbf{A}^\text{op} = \langle A,\top,\bot,\wedge,\vee,\neg\...
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Is there standard mathematical terminology for structures that come in one-to-one pairs?
Background / concrete example of what I'm asking about:
Given a set $X$ and a set of "open" sets $\mathcal T\subseteq\mathcal P(X)$ satisfying the open-set axioms (of topological spaces), we ...
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Annihilator of $H^p$ is isometrically isomorphic to $H_{0}^{q}$
For, $1\le p < \infty$, let $X=L^p[-\pi,\pi]$ and define
$$Y := H^p = \left\{x \in L^p : \widehat{x}(n) = \frac{1}{2\pi}\int_{-\pi}^{\pi} x(s)e^{ins}\ \mathrm ds = 0, n=-1,-2,\ldots\right\}$$
$$H_{...
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Matrix representation of adjoint & co-adjoint orbit of $so(3)$
So I am trying to find the co-adjoint orbits of the lie algebra $so(3)^*$ from this example but I am stuck with a very trivial linear algebra property
now I found the adjoint orbits and I know the ...
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486
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Prove that if $\phi$ is injective then the dual function $\phi^{*}$ is surjective
V,W finite dimensional K-vector spaces.
$\phi:V \rightarrow W$ linear map
I have proven that if $\phi$ is surjective then $\phi^{*}$ is injective. Now I have to prove that if $\phi$ is injective then $...
user900336
2
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1
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148
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Derivation of Dual Curve for Parametric Equations
I am studying projective geometry and am stuck on understanding the following:
For a parametric curve
$$
x = x(t), y = y(t)
$$
the dual curve is given by
$$
X=\frac{-y′}{xy′-yx'},
Y=\frac{x′}{xy′−yx′}
...
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41
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Dual linear map
Is it possible to determine the dual linear map base independently?
Say $V = P_3(x)$ is the vector space of polynomials up to grade 3 and $W = P_2(x)$.
Be $T: V \to W$, $p \mapsto p'$,
what is the ...
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Proving that the curvature of the induced connection satisfies $k_{\nabla^{*}}(X,Y) = k_{\nabla}(X, Y)^{*}$.
I am trying to make Exercise 19 of these lecture notes.
I'll briefly summarize the question below.
Exercise Let $V$ be a dinite dimensional vector space and $A \in \mathrm{End}(V)$. We consider
$$
A^*...
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Is that true that for every linear transformation $\phi : V^* \to W^*$ there is a linear transformation $\psi: W \to V$ such that $\psi^* = \phi$?
$V$ and W are finitely dimensional linear spaces over the field $K$. Is that true that for every linear transformation $\phi : V^* \to W^*$ there is a linear transformation $\psi: W \to V$ such that $\...
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Give example(s) of elements that inside torsion part of cohomology group after quotient
For a manifold $X$, its (integer-valued) cohomology group $H^k(X,\mathbb Z)=Z^k(X,\mathbb Z)/B^k(X,\mathbb Z)$$=\text{Ker}_{\partial_k^*}/\text{Im}_{\partial^*_{k-1}}~$ is construct from the chain ...
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proof of surjectivity of dualT implies injectivity of T and conversely [closed]
This is from Axler's Linear Algebra book page 107 and 108. If $T$ is a linear map from $V$ to $W$, and $T^*$ is a dual map from $W^*$ to $V^*$,
Why is range $T$ = $W$ (surjectivity of $T$) a necessary ...
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Definition of composition used in Axler's proof 3.101, Algebraic Properties of Dual Maps
My question is regarding Axler's proof of this Algebraic Property of dual maps:
$(ST)' = T'S'$ for $T\in\mathcal{L}(U,V), S\in\mathcal{L}(V,W)$ where $T',S'$ represents the dual map.
In the last step ...
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Is rank$(T)$=rank$(T^*)$ even if $T:V\to W$ and $W$ not finite dimensional?
Suppose $T:V\to W$ is a linear map and $V$ is finite dimensional and $W$ is not necessarily finite dimensional. Is it still true that $Rank(T)=Rank(T^*)?$ Where $T^*$ is the dual map of $T.$
I know ...
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Show that $c=\det f$ with $f: V \to V$ is a linear transformation.
(Spivak 4-2.p.g 84). if $f: V \to V$ is a linear transformation and $\dim V=n$, then $f^\ast : \bigwedge^n(V) \to \bigwedge^n(V) $ must be multiplication by some constant $c.$ Show that that $c=\det ...
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Is the equivalent function between isomorphic domains called a dual?
Is the equivalent function between isomorphic domains called a dual or something else? If you can also provide citations or examples of mathematicians using the terminology then that would be great.
...
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Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{*} $ is a projection operator in $ E^{*} $.
Suppose $ \pi: E \rightarrow E $ and $ \pi^{*}: E^{*} \rightarrow E^{*} $ are dual mappings. Assume that $ \pi $ is a projection operator in $ E $. Prove that $ \pi^{*} $ is a projection operator in $ ...
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What is the dual of the space $V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$ with $K\subset\mathbb R^n$ compact?
(1) Could anyone tell me how to find a Dual space of the following space of continuous functions of the following form? And dual maps?
$V=\{f: K\to \mathbb R^n, f(x)=Ax+b\}$, where $K$ is a compact ...
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Proving $\operatorname{coker}(f^*) \cong (\ker f)^*$ for a linear map $f$
Let $f: V \to W$ be linear and $V, W$ be vector spaces of finite dimension. I want to show that the cokernel, defined by $\operatorname{coker}(f^*) := V^* / \operatorname{im}(f^*)$, is isomorphic to $(...
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Is it true that $\dim({\rm im}(f))=\dim({\rm im}(f^{*}))$?
I have a question regarding dimensions of finite vectors spaces.
Let $f$ be a linear map between two vector spaces $f:E_{1}\longrightarrow E_{2}$ with dimensions $m$ and $n$ respectively in a field $\...
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Prove that $\exists f\in V^*$ and $w\in W$ s.t. $A(v)=f(v)w\;\forall v\in V$
Let $V,W$ be (not necessarily finite-dimensional) vector spaces over the field $\mathbb F$ and $A\in L(V,W)$ s.t. $\operatorname{rank} A=1$. Prove that $\exists f\in V^*$ and $w\in W$ s.t.
$$A(v)=f(...
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Write $T'(\phi_1)$ and $T'(\phi_2)$. as a linear combination of $\omega_1,\omega_2,\omega_3$
Define $T : \mathbb R^3 \rightarrow \mathbb R^2~|~T(x,y,z)=(4x+5y+6z, ~~~7x+8y+9z).$ Suppose $\phi_1,\phi_2$ denote the dual basis of the standard basis of $\mathbb R^2$ and $\omega_1,\omega_2,\...
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Show that $(T\circ S)'=S'\circ T'$
Question: Let $S:U\rightarrow V$ and $T:V\rightarrow W$ be linear mappings with S' and T' their transposes. Show that $(T\circ S)'=S'\circ T'$
My approach: Let dimensions of $U,V$ and $W$ are $m,n$ ...
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Finding preimage of point for isogenies between elliptic curves
Let's say one has an isogeny $\alpha:E_1\to E_2$ between two elliptic curves, and that $\ker\alpha$ is known. If there is a point $S_2\in E_2$, is there an efficient way to find its preimage $S_1\in ...
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A question about the dual of super vector space
Let $V$ be a vector space over a field $K$. Denote the dual of $V$ by $V^{*}$, that is $V^*=Hom_K(V,K)$. Suppose there is a morphism $\alpha: V \rightarrow V$. Then we know $\alpha$ induces a morphism ...
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Duality: dual maps and linear maps
Suppose we have $V$ and $W$ be $2$ $K$-vectorspaces and $f:V\rightarrow W$ is a linear map then:
For all $\varphi\in W^\ast$ one has $\varphi\circ f\in V^*$
The map $f^\ast:W^\ast\rightarrow V^\ast:\...
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1
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Dual map and solving linear equation
Let $f:V\to W$ be a linear transformation from a vector space $V$ to a vector space $W$. Suppose that $b\in W$ satisfies $\phi(b)=0$ for all $\phi\in\ker(f^*)$. Show that there exists $x\in V$ such ...
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Proving that $\phi (T) = T^*$ is an isomorphism between vector spaces
I am tasked with the following:
I am thus tasked with proving:
$1)$ $\phi(T)$ is linear, so that it respects closure under scalar multiplication and addition.
$2)$ $\phi(T)$ is a bijection.
I only ...
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If we identify $V$ and $U$ with their canonical images in $V^{**}$ and $U^{**}$ prove that the restriction of $T^{**}$ to $V$ coincides with $T$. [duplicate]
Let $T : V \rightarrow U$ be a bounded map between two normed spaces. Let $T^* : U^* \rightarrow V^*$ be defined by $T^*(f) = f\circ T$ for all $f\in U^*$ (the adjoint map).
My Question:
If we ...
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2
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Annihilator of subspace in terms of a set difference?
I am working through Linear Algebra Done Right by Axler, and I reached the chapter where Null(T'), the Null space of the dual of a linear map, is explored. I attempted to find the answer myself, and ...
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In construction of an inner product that maps from a vector space to a dual, what makes the map considered to be natural? [duplicate]
My question is in reference to the derivation below from the physics site. It shows how the metric tensor raises and lowers indices. I cut it off halfway through because I didn't think the full ...
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Sheldon Axler 3.109 : How to interpret " range T' = $(null\;T)^0$ "?
Excerpt from text:
3.109 The range of T'
Suppose V and W are finite-dimensional and T $\in$ L(V,W). Then
range T' = $(null\;T)^0$
Proof
First suppose $\phi$ $\in$ range T. Thus there exists $\psi$ $\...
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249
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My attempts to show the dual map is isometric.
Theorem: Let $X$ and $Y$ be normed spaces such that $X\cong Y$. Let $\phi:X\rightarrow Y$ be an isometric isomorphism. Then the dual map ${\phi}^*:{Y}^*\rightarrow{X}^{*},\lambda\mapsto\lambda\circ\...
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Prove: If $y_0,...,y_n$ are pairwise different real numbers, then the vectors $f_{y_0},...,f_{y_n}$ form a basis of the dual space $V^*$
could you help me with this task in linear-algebra? I do not know what to do to prove the two following statements in (i) and (ii). I would appreciate it, if you would explain to me the solution in ...
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Any connection between the adjoint map that has determinant $det(\phi)^{(n-1)}$, and the adjoint map that has determinant $ det\phi$?
Is there any connection between the adjoint mapping that is introduced
while studying the matrices, and the adjoint mapping that is
introduced while studying inner product spaces ?
I mean, for ...
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1
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How to prove that every dual linear operator of an operator on $L_2(\mathbb{R})$ shares its eigenvalues with its dual operator
I'm trying to prove that for given two dual maps $A : H \to H$ and $A^* : H^* \to H^*$ where $H = L_2(\mathbb{R})$, the set of all eigenvalues of $A$ is equal to the set of all eigenvalues of $A^*$.
...
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