Questions tagged [dual-maps]

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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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2answers
102 views

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 ...
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Bilinear Maps and their relationships with dual bases

I have a theorem without proof. I have searched many books and tried on myself, but i still dont have the solution. Let M and N F-vector spaces, T be a base of M ,S be a base of N such that dimension ...
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1answer
84 views

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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1answer
52 views

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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2answers
76 views

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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45 views

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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83 views

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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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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38 views

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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80 views

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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1answer
143 views

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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59 views

Prove that $\Omega(x^{*^1}, \ldots, x^{*^n}; x_1, \ldots, x_n) = \phi (x^{*^1}, \ldots, x^{*^n}) \Delta(x_1, \ldots, x_n)$

In the book Linear Algebra by Werner Greub, at page $112$, it is given that, Let $E^*, E$ be a pair of dual vector spaces and $\Delta^* \not = 0, \Delta \not = 0$ be determinant functions in $E^*...