Questions tagged [vector-space-isomorphism]

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

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62 views

Prove that $\mathrm{Ker}(T - \lambda I_V)^n = {0}$ [closed]

I´m having trouble proving this statement, I already tried induction but I failed miserably. Let $V$ a $K$-vectorial space and $T: V\to V$ endomorphism. Let $\lambda\in K$ that it's not a eigenvalue ...
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2answers
64 views

Isomorphism Definition - Error in Hoffman Kunze?

I am Referring to the book Linear Algebra by Hoffman and Kunze $2$e, page $84$ section $3.3$. It defines Isomorphism as follows: If $V$ and $W$ are vector spaces over a field $\mathbb{F}$, then any ...
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2answers
101 views

$\Bbb{R}^n$ and $\Bbb{R}$ are isomorphic as vector spaces over $\Bbb{Q}$.

Show that $\Bbb{R}^n$ and $\Bbb{R}$ are isomorphic as vector spaces over $\Bbb{Q}$. My attempt: Let $c=$ cardinality of $\Bbb{R}$. Then we know that $\Bbb{R}$ over $\Bbb{Q}$ has basis of cardinality ...
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1answer
45 views

Proving a linear map is surjective

Suppose $V_1, \dots, V_m$ are vector spaces. Prove that $\mathcal{L}(V_1 \times \dots \times V_m, W)$ is isomorphic to $\mathcal{L}(V_1, W) \times \dots \times \mathcal{L}(V_m, W).$ (Note that $V_{i}$'...
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1answer
36 views

Homomorphic Image of ideal in Lie algebras

I try to prove the following theorem which given without proof here On Prime Ideals of Lie Algebras Theorem: Let $L$ and $L^{\prime}$ be Lie algebras and let $f: L \rightarrow L^{\prime}$ be a ...
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1answer
34 views

Question on showing invertibility of function $\Phi: \mathcal{L}(V,W) \to M_{m \times n}(F)$ in regards to establishing an isomorphism.

My question comes from a proof of establishing that the function $\Phi: \mathcal{L}(V,W) \to M_{m\times n}(F)$ is an isomorphism. The statement of the theorem comes from Linear Algebra by Friedberg, ...
1
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1answer
41 views

Do properties in linear algebra proved by using matrix transformations hold true irrespective of the choice of the bases for the vector spaces?

Let us say I am required to prove that V (dimension $= n$) and $\Bbb{R} ^ n$ are isomorphic and have chosen the matrix representation way of doing this. Assume a linear transformation of $T :V \...
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1answer
56 views

Isomorphism between $\bigwedge^2\mathbb R^3$ and the dual of $\mathfrak{so}(3)$

I need to show that the map $L:\bigwedge^2\mathbb{R}^3\to \mathfrak{so}(3)^*$ defined by $L(x\wedge y)\Omega=\langle \Omega x,y\rangle$ for $\Omega\in \mathfrak{so}(3)$ is an isomorphism. This map is ...
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1answer
43 views

If $u$ is an isomorphism of $E$ into $F$, prove that to every $n$ there is $m$ such that $u^{-1}(F_n) \subset E_m$

Let $E$, $F$ be two LF-spaces (more details see here or here), $\{E_m\}$, $\{F_n\}$ $(m, n = 1, 2 .... )$ two sequences of definition of $E$ and $F$, respectively. If $u$ is an isomorphism of $E$ into ...
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1answer
43 views

Do theorems concerning isomorphisms that apply to Vector spaces also apply to subspaces?

In chapter 2.4 of Friedberg's text, no mention of subspaces is used in the theorems that he presents concerning isomorphisms. Unfortunately for me, exercise 17 of this chapter involves subspaces. Here'...
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1answer
64 views

About the existence of an isomorphism of algebras between the algebras of $\mathcal{C}^0$ and $\mathcal{C}^1$ functions from $[0, 1]$ to $\mathbb{R}$

Let $\mathcal{C}^0 ( [0, 1], \mathbb{R})$ denote the set of continuous functions from $[0, 1]$ to $\mathbb{R}$ and $\mathcal{C}^1 ( [0, 1], \mathbb{R})$ denote the set of class $\mathcal{C}^1$ ...
3
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2answers
91 views

Let $V$ and $W$ be finite dimensional vector spaces over the field $F$. Prove that $V$ is isomorphic to $W$ iff $dimV=dimW$.

Problem Let $V$ and $W$ be finite dimensional vector spaces over the field $F$. Prove that $V$ is isomorphic to $W$ iff $\operatorname{dim}V=\operatorname{dim}W$. \operatorname{dim} Attempt $\...
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1answer
31 views

Show that $\Bbb{F}^{m×n}$ is isomorphic to $\Bbb{F}^{mn}$.

Problem-Show that $\Bbb{F}^{m×n}$ is isomorphic to $\Bbb{F}^{mn}$. Attempt-Deffine a mapping $T:\Bbb{F}^{m×n}\rightarrow \Bbb{F}^{mn}$ by $T(E_{ij})=e_{n(i-1)+j}$ ,where $1≤i≤m$ and $1≤j≤n$. I assume $...
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1answer
23 views

$L_p(\mu,X)$ is isometrically isomorphic to $\ell_p(X)$

In the book Banach Spaces of Vector-Valued Functions the authors present a demonstration for Proposition 1.6.4, pages 32-33: Let $1\le p \le + \infty$, if $(\Omega,\Sigma,\mu)$ is a $\sigma$-finite ...
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1answer
53 views

Isomorphism by restricting surjective linear mapping between finite dimensional spaces, and quotient space by null space

In Axler's book Linear Algebra Done Right, Section 3D, Exercise 8: Suppose $V$ is finite-dimensional and $T:V\rightarrow W$ is a surjective linear map of $V$ onto $W$. Prove that there is a subspace $...
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0answers
20 views

Why two isomorphic vector space are essentially the same?s [duplicate]

I read the Linear Maps chapter on Sheldon Axler's Linear Algebra Done Right book and he says this: "Think of an isomorphism $T: V \to W$ as relabeling $v \in V$ as $Tv \in W$. This viewpoint ...
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6answers
339 views

Intuition behind Isomorphic spaces “Being the Same”

I know that isomorphic spaces are treated as the same. But why is it so. Like $R^2$ and the set of all ${(x, y, 0) }$ are isomorphic but the " same " vectors in the two spaces are actually ...
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1answer
28 views

An isomorphism between two normed vector spaces with the same finite dimension is an homeomorphism

I'm trying a simple proof of this fact: An isomorphism between two normed vector spaces with the same finite dimension is an homeomorphism. I've tried in this way (everything seems to be ok, I ask ...
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2answers
59 views

Are orthogonal operators always isomorphisms?

I need to show the following: Let $V$ be a finite dimensional vector space with inner product, if $T$ is orthogonal, show that T is injective and surjective I think it is injective because T ...
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3answers
61 views

Is it true that $T$ is orthogonal if and only if $T$ is isomorphism?

I want to prove the following: Let $V$ be a finite dimensional vector space with inner product, then $T$ is orthogonal if and only $T$ is an isomorphism I think the sufficiency could be true because ...
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2answers
55 views

Show that the statements about maps are equivalent

Let $\mathbb{K}$ be a field and $V,W,X$ $\mathbb{K}$-vector spaces. Let $\phi:V\rightarrow W$, $\psi:W\rightarrow X$ be linear maps and let $\phi$ be injective and let $\psi$ be surjective. I want ...
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0answers
21 views

Isomorphism between tensor product of Hilbert spaces

While I am familiar with the isomorphism between finite-dimensional vector spaces with the same dimension and isometric isomorphism between Hilbert spaces, I realize I am getting confused by $n$-th ...
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1answer
47 views

I don't understand the Rank–nullity theorem..

$$\dim(U) = \dim (\ker\phi) + \dim(\text{im }\phi)$$ Hey all I'm just a little confused about the above mentioned theorem; namely, how does one measure the dimensions of a kernel? A kernel is ...
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1answer
32 views

Is $f - 3I$ an isomorphism if $f$ is orthogonal?

Decide if the following statement is true or false by briefly justify the answer. Let $(V, \phi)$ be a real Euclidean space of dimension $n$, and let $f: V \to V$ be an orthogonal operator. ...
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3answers
89 views

$T(p(x)) = p'(x)$ is an isomorphism [closed]

I proved that $T$ is Injective, how can I prove that $T$ is Surjective? Thought about using induction
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1answer
80 views

How do I determine whether or not an isomorphism $T:V\to W$ is a canonical isomorphism?

Roughly speaking, an isomorphism $T:V\to W$ between vectors spaces $V$ and $W$ is canonical if it can be defined without reference to a base. Facts that are widely known are that if $V$ is an ...
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1answer
27 views

Proof verification: $V \cong \mathscr{L}(\mathbb{F}, V)$

Let $V$ a vector space over $\mathbb{F}$ and let the vector space $\mathscr{L}(\mathbb{F},V) = \{f: \mathbb{F} \to V: f$ is linear $\}$. I wanna show that $V \cong \mathscr{L}(\mathbb{F},V)$. My ...
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0answers
11 views

Isomorphism between $\mathcal{l}_p(E) '$ and $\mathcal{l}_q(E')$

Let $1 \leq p < \infty $, $1/p+1/q=1$ and $E$ a Banach space. Define $\mathcal{l}_p(E) = \{ (x_n)_{n=1}^{\infty}\,: \, x_n \in E\,\,\, \forall\,\, n \in \mathbb{N}\,\,\, \mbox{e}\,\,\, || (x_n)_{n=...
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1answer
44 views

Difference between $T \colon V \to W $ and $ T \colon V \to V $ for $\dim(V) = \dim(W)$. [closed]

My professor proved rank-nullity theorem, and after that, he gave two examples for the transformations $T \colon V \to W $ and $ T \colon V \to V $ with $\dim(V) = \dim(W)$. He said something about ...
3
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1answer
69 views

Dual Spaces Isomorphism

So I have to answer the above question as part of homework. I am completely confused by the idea of dual spaces; here, where does the (x,y) term come in? I don't even know how the isomorphism exists ...
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2answers
64 views

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 $(...
0
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1answer
16 views

If the derivative of a continuously differentiable functions between Banach spaces at a point is an isomorphism, is it so also in a neighborhood?

Let $X$ be a Banach space, $U$ an open subset of $X$ and $f:U\to X$ a Fréchet continuously differentiable function. Suppose that, for a point $p\in U$, $Df(p):X\to X$ is an isomorphism. Does there ...
0
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1answer
47 views

Isomorphism Theorems for Lie Algebras

So I stumbled across the isomorphism theorems for Lie Algebras which are as follows: Let $L_1$ and $L_2$ be Lie algebras, $\phi$ : $L_1$ $\rightarrow$ $L_2$ a homomorphism of Lie algebras and $I$,$J$ ...
1
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2answers
46 views

How do you prove this linear transformation $ F \in L(M_{2x2},P_{2})$ is an isomorphism?(verify my solution)

Prove that $$F\begin{bmatrix}a&b\\c&0\end{bmatrix}=ax^2+(a+bx)+a+b+c$$ is an isomorphism. Context: Elementary Linear Algebra Course. Ok, here is what I ve tried by myself. Please, verify my ...
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1answer
26 views

Showing that two Hilbert spaces are isomorphic

I have to show that two Hilbert spaces are isomorphic using the definition professor gave us: Two Hilbert spaces are isomorphic if there exists linear transformation $A: H_1 \rightarrow H_2$ such ...
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0answers
27 views

Is the dual space of a vector space V isomorphic to the space of all linear projections of V onto a 1 dimensional subspace?

It seems like all projections of a vector $v \in V$ to a 1-dimensional subspace represents a method of mapping $v$ to the base field $\mathbb{F}$. Is it necessarily isomorphic? And would any ...
3
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1answer
54 views

All $n$ dimensional real inner product spaces are isomorphic to $\mathbb{R}^n$

I spent a while stuck on this, so now I really want to know if I found a correct solution: Prove: All $n$-dimensional real inner product spaces are isomorphic to the $n$-dimensional Euclidean space. [...
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1answer
51 views

Suppose that $\mathcal{B}$ is a basis for $V$. Prove that $T$ is an isomorphism if and only if $T(\mathcal{B})$ is a basis for $W$.

Let $V$ and $W$ be finite-dimensional vector spaces, and let $T: V \rightarrow W$ be a linear transformation. Suppose that $\mathcal{B}$ is a basis for $V$. Prove that $T$ is an isomorphism if and ...
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1answer
20 views

If $W \approx W'$ and $V \approx V'$ then $L(W,V) \approx L(W',V')$

Exercise: Let's $W,W',V,V'$ vector spaces over a field $F$, with $\dim W =n$ and $\dim V = n$. Suppose that $W \approx W'$ and $V \approx V'$ then $L(W,V) \approx L(W',V)$ (Show an isomorphism). ...
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1answer
52 views

$V$ and $W$ are finite-dimensional isomorphic vector spaces, where we have a basis for $W$

Say I have two finite-dimensional vector spaces $V$ and $W$. I know that the two are isomorphic. I also know a basis for $W$. I claim a basis for $V$. In order to show that this indeed is a basis ...
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1answer
62 views

Linear algebra - Prove that for any isomorphism there is an “identity basis(?)”

I'd appreciate if you could help me with this question. What I thought about so far is showing that for any basis $B$, we know that $[T]_B$ is invertible so there are $k$ elementary matrices such that ...
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1answer
37 views

Inner product gives vector space isomorphism with dual space

Let $V$ be a finite dimensional vector space over the field $\mathbb{K}$, then $V$ is isomorphic to its dual $V^*$. To see this let $B=\{e_1,...e_n\}$ is a basis for $V$. Then any vector $v \in V$ ...
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1answer
19 views

Question regarding Isomorphism and Isomorphic vector spaces

I am self reading Linear Algebra Done Wrong by Treil. In that, I am currently studying Isomorphism and find it hard to understand what it really means. An invertible linear transformation $A: V\to ...
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1answer
78 views

Rank nullity theorem and domain(T) being isomorphic to direct sum of kernel(T) and image(T)

Let $T : \mathbb{R}^n \xrightarrow{} \mathbb{R}^n$. The domain of $T$ is finite dimensional $\mathbb{R}^n$, so by rank nullity theorem we have $n=$ dim(ker($T$)) + dim(im($T$)). Although I do not ...
0
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1answer
49 views

Left invariant vector field and Tangent space at $e$ on a Lie group $G$

I have to say sorry about my English which is not my native language. So you can asking me for the ambiguous concept of what I wrote. Notation: $(G,\cdot)$is a Lie group, $\mathscr{L}$ is left ...
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2answers
213 views

Isomorphism mapping from $\mathbb R^3$ to $\mathbb P_2(\mathbb R)$

Can you give an example of an isomorphism mapping from $\mathbb R^3 \to \mathbb P_2(\mathbb R)$(degree-2 polynomials)? I understand that to show isomorphism you can show both injectivity and ...
0
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1answer
45 views

Isomorphism between $\mathbb{gl}(V)$ and $\mathbb{gl}(n,\mathbb{F})$ if $V\cong \mathbb{F}^n$

This is probably a standard example, but I couldn't really find it anywehere and I'm unsure if I understood it correctly... Take any finite dimensional vectorspace $V\cong \mathbb{F}^n$, then the Lie ...
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0answers
42 views

Detail in the Fredholm alternative: necessity to prove $\operatorname{codim}\operatorname{Im}(I-K) = \operatorname{dim}\operatorname{Ker}(I-K)$

For storytelling purposes, let me recall the theorem and the different steps of the proof. The question is in boldface after 2. followed by an "almost answer" in comments... Let $\big(E,\lVert\,\cdot\...
3
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2answers
57 views

What is the connection between the fundamental subspaces and finite-dimensional vector spaces being isomorphic to $\mathbb R^n$

I know that if $V$ is a finite dimensional vector space of dim $n$, then $V$ is isomorphic to $\mathbb{R}^n$. We know this is true because if dim $V = n$, there is some basis $\mathbf{v}_1, \mathbf{v}...
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0answers
18 views

Direct sum of HIlbert spaces isomorphic to…

I need to clear my mind in order to solve some excersices: I need to proof that $\mathbb{C}^{n_1} \oplus \mathbb{C}^{n_2} \cong \mathbb{C}^{n_1+n_2}$ where $\mathbb{C}^{n_1}$ and $\mathbb{C}^{n_1}$ ...

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