Questions tagged [vector-space-isomorphism]

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

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3
votes
2answers
46 views

$F'$ and $E'$ are isomorphic isometrically.

Let $F$ be a dense subspace of the normed space $E$. Prove that $F'$ and $E'$ are isomorphic isometrically. At first I was trying to define a function to check the isomorphism and then show that it ...
1
vote
1answer
45 views

Show $T: \mathcal{P}_n \rightarrow \mathcal{P}_n$ defined by $T(p(x)) = p(x) + p'(x)$ is an isomorphism

Show $T: \mathcal{P}_n \rightarrow \mathcal{P}_n$ defined by $T(p(x)) = p(x) + p'(x)$ is an isomorphism Let $c_0 + c_1x+c_2x^2 + ... + c_nx^n \in \mathcal{P}_n$ Suppose $T(c_0 + c_1x+c_2x^2 + ... + ...
1
vote
0answers
48 views

Show a vector space has no countable basis

The question is: Prove that there is no isomorphism $T:W\rightarrow V=\mathbb{R}^\mathbb{N}$, hence deduce that $V$ has no countable basis. I think I know how to show $V$ has no countable basis: ...
2
votes
0answers
33 views

No isomorphism between set of finite and infinite sequences

I have this in my notes and now I can't remember why that's the case: If $V$ is the set of finite sequences (i.e. finitely many non-zero entries) and $V'$ is the set of infinite sequences, then $V$ is ...
0
votes
0answers
32 views

$T$ is an isomorphism if and only if there are $c,k>0$ such that $c||x|| \leq ||T(x)|| \leq k||x||$.

Let $T : E \longrightarrow F$ a linear bijective operator between normed spaces, then $T$ is an isomorphism if and only if there are $c,k>0$ such that $$c||x|| \leq ||T(x)|| \leq k||x||,$$ for all $...
0
votes
1answer
33 views

Regular matrix and linear transformation map

Let $A$ be a matrix with dimension $n\times n$ , $B$ is a matrix with dimension $m\times m$ then if $f : R \to R$ where $R$ is the set of matrix with dimension $m\times n$ and $f(C) = ACB$ is ...
1
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0answers
63 views

Is C(K) a subalgebra in its bidual?

Let $C(K)$ be the Banach algebra of real-valued continuous functions on a compact Hausdorff space $K$. It is known that the bidual of $C(K)$ is again a $C(\Omega)$ space, for some compact Hausdorff ...
0
votes
1answer
28 views

Quotient space projection

Let $U\subset\Bbb R^2$ be a line through the origin. Describe the quotient space $\Bbb R^2/U$. Then let $V\subset \Bbb R^2$ be a different line that goes through the origin. Consider the projection $$\...
0
votes
0answers
43 views

Let $A \subseteq \mathbb R^m$, $B \subseteq \mathbb R^n$, and $f:A \to B$ be a diffeomorphism. Then $m = n$

In proving the dimension of a manifold is well-defined, I have come across this lemma. Could you check if my proof is fine or if it contains subtle logical mistakes? Let $A \subseteq \mathbb R^m$, $B ...
1
vote
0answers
25 views

Understanding solutions given by transposition

I'm studying solution given by transposition and i would like to clear somethings. Assume we have the following, let $T < \infty$ and $Q=\Omega \times] 0, T[$, let $\varphi$ given in a Hilbert ...
0
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0answers
38 views

Show that the vector space $V$ over Field $R$ with a basis $B$ and vector space $\text{Map}[B,R]$ are isomorphic.

Question: Show that the vector space $V$ over Field $R$ with a basis $B$ and vector space $\text{Map}[B,R]$ are isomorphic. Show $V \cong \text{Map}[B,R]$. ($\;\text{Map}[B,R] := \{f \in \text{Map}[...
0
votes
0answers
27 views

Existence of linear map.

Let $V$ be a finite dimensional vector space. Let $v_1 , v_2 \in V$ and $v_1 \neq 0$. I need to show that there exists a linear map $S$ such that $S(v_1) = v_2$. I thought of using the fact that since ...
0
votes
1answer
26 views

Why is addition not commutative under PM's notion of relation number?

Quoting Bertrand Russell's "The Principles of Mathematics" p468 ยง299: It is worth while to repeat the definitions of general notions involved in terms of what may be called relation-...
0
votes
1answer
52 views

Is a quotient space isomorphic to its orthogonal

The quotient space of a normed space $(X, \|\cdot\|_X)$ by a closed subspace $M\subset X$ is the normed space $(X/M, \|\cdot\|_{X/M})$ defined as follows \begin{align*} X/M= \{\dot{x}=x+M: x\in X\}\...
-1
votes
2answers
57 views

Why is dimension an invariant between finite-dimensional isomorphic vector spaces?

Finite-dimensional vector spaces, Paul R. Halmos, reprint of 2nd edition, paragraph 9, "Isomorphism": Definition. Two vector spaces $\cal{U}$ and $\cal{V}$ (over the same field) are ...
0
votes
2answers
52 views

How should I read/interpret this notation?

The notation used is $\boldsymbol{v} \mapsto [\boldsymbol{v}]_B$, I've read it before but I don't really understand. It appears in the following proof for context: Two finite-dimensional vector ...
0
votes
1answer
48 views

two norms are equivalent

I have a homework question that ask to prove two parts and I did one part and not sure how to prove the second part. The question says: Let $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ be two normed vector ...
1
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0answers
20 views

Proving that an isomorphism from a vector space $X$ to $X'$ using symmetric bilinear forms is onto

Here's the problem: Let $f$ be a symmetric bilinear form over an n-dimensional vector space $X$, and $f$ is non-degenerate (for all nonzero $x \in X$, there exists $y \in X$ so $f(x,y) \neq 0$). Prove ...
2
votes
2answers
71 views

Showing a linear transformation $T\colon V^{*}\to\mathbb R^{m}$ is surjective

Let $V$ be a finite dimension vector space and let $V^{*}=\operatorname{Hom}(V,\mathbb R)$ be its dual space. Let $v_{1},\ldots,v_{m}\in V$, and let $T:V^{*}\to\mathbb R^{m}, \ \ T(\phi)=\pmatrix{\phi(...
3
votes
1answer
49 views

Matrix spaces A,B with the same dimension implying B=QAP?

I have a conjecture: If two finite-dimensional matrix spaces $\mathcal{A},\mathcal{B}\subseteq \mathrm{M}(n,\mathbb{F})$ have the same dimension, then there would exist $P,Q\in\operatorname{GL}(n,\...
0
votes
1answer
71 views

Doubt about $\mathbb C\left[ X\times Y \right]$ and $\mathbb C\left[ X\right]\times\mathbb C\left[ Y\right]$ being isomorphic.

Here is my doubt. In case it's not clear what the notation in the title means, I'll report it here: $\begin{equation*}\mathbb C\left[ X\right]\colon =\left\{ \sum\limits_{x\in X} a_x\cdot e_x\:\middle\...
2
votes
2answers
53 views

Prove that Diag$_n(\mathbb{R})$ is isomorphic to $\mathbb{R}^n$

Prove that each of the following spaces are isomorphic to $\mathbb{R}^d$ for some $d$ and compute this number. (a) Diag$_n(\mathbb{R})$, that is diagonal matrices of order $n$ (b) UT$_n(\mathbb{R})$, ...
1
vote
2answers
80 views

Can an arbitrary vector space over $\mathbb{F}$ be written as $\mathbb{F}^X$ for some set $X$?

Each tuple $(x_1, \ldots,x_n) \in \mathbb{F}^n$ can be thought of as a function $f: \{1,\ldots,n\} \rightarrow \mathbb{F}$ where $f(i) = x_i$ for $i \in \{1,\ldots,n\}$. Similarly any $f: \{1,\ldots,n\...
0
votes
1answer
288 views

Show that $V$ and $\mathcal{L} (\mathbb{F},V)$ are isomorphic vector spaces.

In LADR by Axler, there are two theorems stated as following: Theorem 1: Two finite-dimensional vector spaces over $\mathbb{F}$ are isomorphic $\iff$ They have the same dimension. Theorem 2: $V$ and $...
0
votes
0answers
67 views

Isomorphism between $V^* \otimes V$ and $\operatorname{End} V$

If $V$ is a finite dimensional vector space and $\operatorname{End} V$ is the set of endomorphisms from $V$ to $V$. Defining a map from $V^*\otimes V\rightarrow \operatorname{End} V$ by sending some ...
0
votes
1answer
48 views

Proving isomorphism between two spaces of the linear transformations [duplicate]

Let $๐‘ˆ$ and $U'$ be vector spaces isomorphic and $V$ and $V'$ be vector spaces isomorphic We have to show that $๐ฟ(๐‘ˆ,๐‘‰)$ and $๐ฟ(๐‘ˆ',๐‘‰')$ are also isomorphic. Where, $๐ฟ(๐‘ˆ,๐‘‰)$ is the space of ...
8
votes
4answers
252 views

Natural isomorphism in linear algebra: is the naturalness asymmetric?

Let $V$ be a finite-dimensional vector space. It is well known that there is a natural isomorphism between $V$ and its double dual $V^{\ast\ast}$ defined by $T(x)(f)=f(x)$ for every $x\in V$ and $f\in ...
0
votes
1answer
96 views

Can $L^p(\mathbb{R})$ and $L^p[0,1]$ be isomorphic?

Let $1\leq p<\infty$. Are the Banach spaces $L^p(\mathbb{R})$ and $L^p[0,1]$ isomorphic? The case $p=\infty$ can be treated by seeing that $\mathbb{R}$ and $(0,1)$ are homeomorphic. Can this be ...
0
votes
2answers
152 views

Isomorphism between dual of quotient and annihilator

While doing exercises I found this particularly surprising result: $(V/W)^* \cong W^0$, where $W^0$ denotes the annihilator of $W$. I wanted to prove this result by giving an explicit linear map (the ...
0
votes
1answer
29 views

if $L:V\to V'$ is a surjective linear aplication, and $W$ is a subspace of $v$, exists isomorphism $\frac{V}{W+\ker{L}}$ and $\frac{V'}{L(W)}$

Let $L\colon V\to V'$ a surjective linear map. Let $W\subseteq V$ a vector subspace. I'm trying to show that exists an isomorphism $$ \frac{V}{W+\ker{L}}\simeq \frac{V'}{L(W)}. $$ I think to define ...
2
votes
4answers
214 views

Prove that 2 vector spaces are isomorphic to each other

Prove that the vector space V $V = \begin{bmatrix} a& 0& 0& 0 \\ b& 0& 0& 0 \\ c& d& e& f \\ \end{bmatrix}$ is isomorphic to the vector space $M_{2,3}$ ...
1
vote
0answers
32 views

Showing the existence of an automorphism $v$ such that $v=f$ in the supplement of $Ker(f)$

Let $f$ be an endomorphism of a vector space $E$ that has a finite dimension. Let $F$ be a sub vector space of $E$ such that $E=F + Ker(f)$ is a direct sum, $Ker(f)$ is the kernel of $f$. Show that ...
3
votes
2answers
155 views

Prove that $T$ is surjective

Let $T:S_2 \to \mathbb{R}_3[x]$ such that $$T\left(\begin{bmatrix} a & b\\ b & c \end{bmatrix}\right)=(a+b+c) + (-a+2c)x+(2a+3b+6c)x^2.$$ Prove that $T$ is surjective. My attempt: Let $w \...
1
vote
1answer
82 views

Dual Space of $\mathbb{R}^n$

I am trying to understand dual spaces. Suppose I have the Euclidean vector space $\mathbb{R}^n$ over the field $\mathbb{R}$. Elements of this space are column vectors $\boldsymbol{x}\in\mathbb{R}^{n \...
0
votes
1answer
66 views

Prove that the map that takes $\mathcal{L}(V,W)$ to $\mathcal{L}(W',V')$ is an isomorphism of $\mathcal{L}(V,W)$ onto $\mathcal{L}(W',V')$

I was reading Axler's Linear Algebra Done Right, and the following question appears as exercise $16$ of chapter $3$, section F: Suppose V and W are finite-dimensional. Prove that the map that takes $\...
1
vote
2answers
186 views

Prove that $V$ is isomorphic to $U \times (V/U)$

I was reading Axler's Linear Algebra Done Right, and the following question appears as exercise $12$ in chapter $3$, section $E$. Suppose $U$ is a subspace of $V$ such that $V/U$ is finite-...
1
vote
1answer
80 views

Prove that $V^n$ and $\mathcal{L}(\mathbf{F}^n,V)$ are isomorphic vector spaces

For $n$ positive integer, define $V^n$ by $V^n=\underbrace{V\times...\times V}_{n \ times}$. Prove that $V^n$ and $\mathcal{L}(\mathbf{F}^n,V)$ are isomorphic vector spaces. I would like to know if my ...
0
votes
0answers
25 views

Reference for integral equation result

Let $T>0$, $X := L^2([0, T])$ and $\Gamma: X \rightarrow X$ be defined as $$ (\Gamma u)(t) := \int^t_0 k(t, s)u(s)~\mathrm{d}s, $$ where $k \in L^2([0, T]^2)$. I am rather certain that $\mathrm{Id} ...
0
votes
0answers
36 views

How to prove isomorphism of $\mathcal{L}(V,W)$ onto $\mathcal{L}(W',V')$

This question concerns problem 3F 16) in Axler's (2005) "Linear Algebra Done Right". Assuming that V and W are finite-dimensional vector spaces: how do I prove that a map that takes $T\in\...
1
vote
1answer
25 views

image of positive real number under field isomorphism

Claim: If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a field isomorphism, then $f(x) >0$ for any $x>0$. I know $f(x)$ can be determined by $f(1)$ the generator of $\mathbb{R}$ but not sure how to ...
1
vote
0answers
54 views

determining the isomorphism.

Let $\mathbb C$ be the field of complex numbers and $\mathbb H$ the algebra of quaternions we want to show that $$\mathbb{C} \otimes_{\mathbb R} \mathbb H \cong_{\mathbb R} M_2(\mathbb C).$$ Observe ...
0
votes
0answers
12 views

Is there an equivalence between a graded vector space and its degree-shifted counterpart?

I am studying graded vector spaces and I have a simple question. Let me denote by $\mathbb Z_2=\mathbb Z\text{mod}2=\{-1,0,+1\}$. Now let's perform a shift by an integer $k$ and get the set $\mathbb ...
1
vote
1answer
27 views

Dual basis example and the isomorphism $V\cong V^{\ast }$

I am trying to understand the dual spaces and working on some problems. Let $V=\mathbb{R}^{3} $ with the base $b=\left \{ (1,0,1),(1,2,1),(0,0,1) \right \}$ find the dual basis $ b^{\ast }$ of the ...
1
vote
1answer
31 views

Isomorphism between vector space of all linear transformations

If $V$ and $W$ are vector spaces, is there an isomorphism between $L(W, V^*)$ and $L(V,W^*)$? If so, how do I find this? ($L(V,W)$ is the vector space of linear transformation from $V$ to $W$). Any ...
4
votes
4answers
242 views

What structure is preserved in a vector space isomorphism?

The definition of an isomorphism is "a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping". In my linear algebra class, an ...
0
votes
0answers
17 views

Isomorphism and Invertible Matrices [duplicate]

I was reading a book of linear algebra and started the chapter of isomorphisms. The book describes them as linear maps that are injective and surjective. Then I realised that that condition is the ...
2
votes
1answer
35 views

Weird condition implies isomorphism between vector spaces?

Let $U$ and $V$ be vector spaces. It can be shown that the existence of atleast one space $W$ such that $\mathscr L(U,W) \cong \mathscr L(V,W)$ is not sufficient for $U$ and $V$ to be isomorphic as, ...
1
vote
1answer
38 views

values of determinants?

Let $V,W$ be finite-dimensional $\mathbb{K}$-vector spaces of dimension $n \gt 0$ and $\varphi:V\to W$ a linear map. a) Prove: for every ordered basis $B$ of $V$ and every ordered basis $C$ of $W$ - $\...
0
votes
1answer
56 views

Isomorphism map between $\frac{\mathbb{Z}_3[x]}{(x^2+x+2)}$ and $ \frac{\mathbb{Z}_3[x]}{(x^2 + 1)} $ [duplicate]

$\frac{\mathbb{F}_3[x]}{(x^2+x+2)} \cong \frac{\mathbb{F}_3[x]}{(x^2 + 1)} $ I know $\frac{\mathbb{F}_3[x]}{(x^2+x+2)} $ and $ \frac{\mathbb{F}_3[x]}{(x^2 + 1)} $ is isomorphic because they are both 2-...
0
votes
1answer
31 views

Connection between vector space isomorphisms and dimensions

Assuming the Axiom of Choice, and therefore assuming that every vector space has a Hamel basis, it isn't too hard to show that if $U$ and $V$ are vector spaces with the same dimension (i.e. their ...

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