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# 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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### 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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### 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 ...
### 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 ...