# Questions tagged [vector-space-isomorphism]

This tag should be used for questions about isomorphisms between vector spaces.

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• 87
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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 ...
• 799
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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 $T^{-1}$...
• 123
1 vote
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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-...
• 300
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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 ...
• 182
257 views

### 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 ...
87 views

### 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 ...
86 views

### 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 ...
99 views

### 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 ...
184 views

### 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 ...
• 197
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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 ...