# Questions tagged [dual-maps]

The tag has no usage guidance.

16 questions
Filter by
Sorted by
Tagged with
59 views

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 ...
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 ...
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 ...
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 $A^*$. ...
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 ...
28 views

### 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 ...
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 ...
71 views

### 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 $m,n$ ...
84 views

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 ...
20 views

### 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 ...
24 views

### 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 ...
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 ...