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Questions tagged [vector-space-isomorphism]

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

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Proof of ABCD Lemma

A standard result in bifurcation theory due to Keller, known as ABCD Lemma, asserts that if $V$ is a Banach space, $M\,:\,V\times \mathbb{R} \to V\times \mathbb{R}$ is such that $$ M:=\begin{pmatrix} ...
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39 views

Showing isomorphism in transformation between matrix and bilinear map spaces

From Serge Lang Linear Algebra: Show that the association $A \rightarrow g_A$ is an isomorphism between the space of $m \times n$ matrices, and the space of bilinear maps of $\mathbb{K}^m \times ...
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1answer
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Regular representation, two viewpoints, the isomorphism

Let $G$ be a group. Let $G^* = \{\phi: G\to \mathbb{C}\}$ be the complex functions defined on $G$. We have the representations $$\rho: G \to GL(G^*), \space \rho(g) = \phi \mapsto (h\mapsto \phi(hg))...
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1answer
35 views

Canonical transformation $T_pV \cong V$?

Apparently there is a natural isomorphism between a vector space and the tangent space of this vector space at one point. I.e., $\forall p \in V$ we have $T_pV \cong V$. I know that the isomorphism is ...
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2answers
196 views

Are two submodules (where one is contained in the other) isomorphic if their quotientmodules are isomorphic?

Let $M$ be an $R$ module and $N_1 \subset N_2$ be submodules of $M$ such that $M / N_1 \cong M / N_2$. Can I know conclude $N_1 \cong N_2$ or even $N_1 = N_2$? I know that a proper submodule can be ...
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Given a smooth map $\sigma$, and a linear isomorphism $K$, is there a smooth map $\tau$ s.t $D(\tau \circ \sigma^{-1}) = K$

Let $(M, \Sigma)$ be a smooth manifold, and $\sigma, \tau$ be two smooth charts defined on the neighbourhood of $x_0 \in M$. Then by definition $$\tau \circ \sigma^{-1}$$ is a diffeomorphism from an ...
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39 views

Are any two isomorphic normed linear spaces homeomorphic?

We know that any two finite-dimensional linear spaces (over a same field) of same dimension is isomorphic. Qn.1 Are any two finite-dimensional normed linear spaces (over a same field) with same ...
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Prove that there exists an ordered basis $\gamma$ for which $[T^*]_\gamma$ has a column of $0$s.

$V$ is an $n$-dimensional vector space over a field $\mathbb{F}$. Assume that $T^*:V\rightarrow V$ is a linear operator on $V$ and $T^*$ is not an isomorphism. Prove that there exists an ordered basis ...
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Quotient of quotients in finite dimensional vector spaces

Suppose we are given filtrations of finite dimensional vectors spaces: $$ B_d\subseteq Z_d\subseteq C_d$$ $$0 \subseteq C_{d,1} \subseteq C_d$$ $$0 \subseteq Z_{d,1} \subseteq Z_d$$ $$0 \subseteq B_{...
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1answer
36 views

Show that $Γ$ is isomorphic to $L(V/W,V')$

Question: let $V$, $V'$ be vector spaces over field $K$ and $W$ be subspace of $V$ then show that $\Gamma = \{T\in L(V,V')\vert\forall w\in W: T(w) = 0\}$ is subspace of $L(V,V')$. Further show that ...
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Proving entire sequence space and the subspace of sequences with finitely many non-zero entries are not isomorphic

Let $\mathbb{R}^{\infty}$ be the vector space of all real-valued sequences (over the field of real numbers), i.e., $$\mathbb{R}^{\infty}:= \{f: \mathbb{N}\rightarrow\mathbb{R}\}.$$ On the other hand, ...
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Sheldon Axler 3.117: How to understand the proof

I have trouble in understanding the proof of 3.117 in Linear Algebra Done Right (Third Edition). The Theorem is the following: 3.117$\quad$ Dimension of range $T$ equals column rank of $\mathcal{M}...
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33 views

Why does $\mathscr{P}_2$ being isomorphic to $R^3$ imply that a constantwise addition of objects in $\mathscr{P}_2$ is an inner product?

Why does saying that $\mathscr{P}_2$ is isomorphic to $R^3$ immediately prove that a constant-wise addition of objects in $\mathscr{P}_2$ is an inner product? What I mean by constant-wise addition ...
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2answers
159 views

Invariants between two isomorphic vector spaces

I have a general question about isomorphisms between vector spaces. From a general point of view, there are common properties (invariants) between two isomorphic structures (e.g., properties about ...
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30 views

Prove that if $f$ is a diffeomorphism than its differential $D_{f}$ is an isomorphism

I want to prove that if $f : U \to V$ , when $U,V \subset R^n$, is a diffeomorphism than its differential $D_{f}$ is an isomorphism. Since $D_{f}$ is a linear transformation, isomorphism means that ...
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How can vector space $V\subsetneq W$ but $V\cong W$, where $V$ and $W$ don't have finite dimensions?

Let $V$ and $W$ two vector spaces. If they have finite dimension, then $V\subset W$ and $V$ and $W$ have same dimension will imply that $V=W$. But I heard that in infinite dimensions, this doesn't ...
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2answers
41 views

Prove that $\Bbb R^n$ and $\mathcal L(\Bbb R,\Bbb R^n)$ are isomorphic.

Is this question asking me to prove there exists an invertible linear map between $\Bbb R^n$ and the set of all linear maps between $\Bbb R$ and $\Bbb R^n$? And if so, how would I do that?
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1answer
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Let $V$ be a finite dimensional real vector space. If $S,T∈L(V,V)$ prove that $ST$ and $TS$ have the same eigenvalues whenever $T$ is an isomorphism

I know the result is true even when T is not an isomorphism but how would I show it if it was one, however.
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Show $T$ is an invertible linear map iff $\operatorname{rank} T = \dim V$

Lemma: Let $W$ be a subspace of $V$. $\dim V= \dim W \iff V=W$. Proof: If $V=W$ then clearly $\dim V=\dim W$. Assume $\dim V=\dim W$. Let $B= \{w_1,...…,w_n\}$ be a basis for $W$ then since $W \...
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Why are $T^k(V)$ and $V^* \otimes… \otimes V^*$ just isomorphic?

In Lee, p.178, it is said that $T^k(V)$ is isomorphic to $V^* \otimes... \otimes V^*$. But is this not stronger? Isn't this an equality? I thought if $v_1,..., v_n$ is a basis of $V$ and $\epsilon^1,.....
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1answer
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Proof or counterexample for isomorphism of group representations

Let $(\pi , V_\pi)$ be an irreducible unitary representation of the (locally compact) group $G$. Let $V_\pi^n = V_\pi \oplus \ldots \oplus V_\pi$ be the $n$-fold direct sum of $V_\pi$ on which we have ...
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1answer
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Isomorphisms and Linear Maps

WTS: $V\simeq$ W $\iff$ $dimV=dimW$ My proof: Assume $T: V \rightarrow W$ is an isomorphism, therefore the kernel for the linear map is the zero vector therefore $dim(ker(T))=0$ and since it is ...
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1answer
48 views

If $T\in B(X,Y)$ and $T$ is bijective is $T^*$ also bijective?

1) This first link seems to provide a proof that seems to work for all normed vector spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? 2) This second ...
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Let $T: V \rightarrow W$ be a vector space isomorphism. Prove that the $T$ maps proper subspaces to proper subspaces.

Let $T: V \rightarrow W$ be a vector space isomorphism. Prove that the $T$ maps proper subspaces to proper subspaces. Let $T:V_1 \rightarrow W_1.$ Prove that if $V_1$ is a proper subspace of $V,$ ...
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2answers
40 views

Show that an operator with certain properties is an isomorphism

Suppose $H$ is an infinite dimensional separable Hilbert space, with $\langle \cdot,\cdot\rangle$ denoting the duality pairing between the dual $H'$ and $H$ and $(\cdot,\cdot)$ the inner product on $...
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2answers
29 views

Linear algebra, proof that it's an isomorphism.

I'm dealing with isomorphisms, and I'm not quite sure of how to formulate this one. Mainly the elements from each function, so I can start to proof that is a linear and a bijective function. Here are ...
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isomorphism of Cartesian product of two vector spaces

$\mathbb R^d\times \mathbb R^d\times\cdots\cdots\times\mathbb R^d$ are said direct product of $n$ copies of vector spaces, which is isomorphic to a vector space of dimension $nd$. Do we send a basis ...
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Banach spaces and isomorphism between dual spaces

Maybe is a silly question, but I have got a doubt about it: Let $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ be two linearly isomorphic Banach spaces. Then we know that their dual spaces are ...
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1answer
38 views

Endomorphism on polynomial vector space $\mathbb{R}_3[x]$

Let $ \mathbb{R}_3[x] $ be the vector space of the polynomials with the degree $ \le 3$. Given the endomorphism on this vector space, $$ T:\mathbb{R}_3[x] \to \mathbb{R}_3[x], T(f)(x) = f(x+1)-f(x), $$...
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How to show isometry of the space through plane?

I am totally new to the isometries of the plane and space. I have to prove that the map $$R(\vec{x})=\vec{x}-2(\vec{x}\bullet\hat{n})\hat{n}$$ from $R^{3}$ to $R^{3}$ is isometry of the space $R^{3}.$ ...
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1answer
37 views

$ \operatorname{Ker}TR = \{0\}$ but $ \operatorname{Ker}T \not= \{0\}$ where $R$ is an isomorphism?

Let $V$ be a finite-dimensional vector space, and let $R: V \rightarrow V$ be an invertible linear transformation. Is there a linear transformation $T: V \rightarrow V$ such that $ \operatorname{Ker}...
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Is $ f(A) = A + 2A^{T} $ an isomorphism of $ \mathbb R^{5,5} $ onto itself?

I have problem with prove or disprove this hypothesis: Is the linear transformation $ f \in L(\mathbb R^{5,5},\mathbb R^{5,5}) $ $$ f(A) = A + 2A^{T} $$ an isomorphism of the space $ \mathbb R^{...
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2answers
292 views

Is $V$ isomorphic to direct sum of subspace $U$ and $V/U$?

Given a vector space $V$ and a subspace $U$ of $V$. $$ V \cong U \oplus(V/U) $$ Does the above equation always hold? Where $\oplus$ is external direct sum. For finite dimensional vector space $V$, ...
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Isomorphism of X to itself. [closed]

Let $(X, +, \bullet)$ and $(X,\tilde{+}, \tilde{\bullet})$ be vector spaces over $\mathbb{R}$, and dim$(X, +, \bullet) =n$ (the set $X$ is the same in both spaces). Show that $(X, +, \bullet)$ is ...
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60 views

Linear map $L \neq O$ having trivial image only at $L^2=L \circ L$

From S.L Linear Algebra: Let $L:ℝ^2 \rightarrow ℝ^2$ be a linear map such that $L \neq O$ but $L^2=L \circ L=O$. Show that there exists a basis $\{A, B\}$ of $ℝ^2$ such that, $L(A)=B$ and $L(B)...
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1answer
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Isomorphism and dimension exercise clarification.

I'd need clarification on one of the statements here below. Given that V is the set of linear operators f: $\mathbb{R}^k \rightarrow \mathbb{R}^n $, and I'm trying to prove that V has the same ...
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1answer
44 views

Show that $T$ is an isomorphism of X in a closed subspace of $Y$.

Let $T \in B(X,Y)=\{T \in \mathcal{L}(X,Y) | T$ is bounded $\}$ with $X$ and $Y$ Banach space such that it exist $\delta >0$ , $\|Tx\| \geq \delta \|x\|$ for all $x \in X$. Show that $T$ is an ...
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1answer
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Finding an explicit isomorphism between $\mathbb R^4 / \ker \ T$ and $\mathbb R^2$

I'm wondering if I have a valid answer to this. It is exactly (e) of the following: I first state that the two vector spaces are isomorphic because they have equal dimension. I then define a ...
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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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Isotropy/little group of $O(n)$

I'm trying to prove that the little group of $O(n)$ acting on a $k$-dimensional subspace of $\mathbb{R}^n$, call it $V$, is $O(k)\times O(n - k)$ due to the Grassmann manifold is isomorphic to $O(n)/(...
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1answer
25 views

Hyperplane is isomorphic to the entire space in infinite dimension

Let be $E$ a vector space of infinite dimension. I have the intuition that for all hyperplanes $F$ of $E$, there is an isomorphism between $F$ and $E$, it suffices to show this is true for an ...
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Schauder basis under isometry.

I am self studying linear functional analysis and am a bit confused about the following problem. It is situated after a chapter on the open mapping and closed graph theorem and in my answer I don't ...
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22 views

Vector Space Isomorphism related to Tensor Product

$V$ and $W$ are finite dimensional vector spaces over $k$. I need a basis free isomorphism between $V^*\otimes_{k} W^*$ and $Bil_{k}(V\times W,k)$. My attempt: We have a bilinear map $V^*\times W^*\...
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1answer
56 views

Isomorphism and Basis

I started reading Linear algebra done wrong and am confused at a theorem that he leaves the proof off to the reader. The Theorem is if A: V -> W is an isomorphism, and v1, v2, ... vn are a basis in V. ...
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1answer
93 views

Associativity of direct sums

Given three vector spaces U, V, and W, which aren't necessarily subspaces of a common vector space, I have to prove that (U $\oplus V) \oplus W \cong U \oplus (V \oplus$ W). I don't even know how I ...
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1answer
87 views

Show linear mapping is isomorphism

The question; show that the linear mapping for which $$ 1 \rightarrow \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, i \rightarrow \begin{bmatrix} i & 0 \\ 0 ...
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147 views

Linear isomorphism beetween finite dimensional spaces

Let $T:(\mathbb R^n,\|.\|_1)\to (\mathbb R^n,\|.\|_2)$ be a linear isomorphism. Then clearly it is a topological isomorphism also. I want to show that $\|T\|\|T^{-1}\|\geq \sqrt{n}$. Here $\|(...
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2answers
48 views

Is there an isomorphism between projections?

Let $V$ a finite dimension vector space and let $T$ be a linear operator such that $T^2 = T$. Suppose there's another linear operator $R$ such that $R^2 = R$. Is there an isomorphism $\rho$ such that $...
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1answer
17 views

Circulant Matrices and Ring of Polinomials

I am having problem to proove that the group of circulant Matrices of size nxn, $C_n$ is isomorphic to $\frac{\mathbb{C}[x]}{<x^n-1>}$