# Questions tagged [vector-space-isomorphism]

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

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### 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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### 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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### $\Bbb{R}^n$ 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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### 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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### 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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### 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, ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $T(p(x)) = p'(x)$ 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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### 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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### 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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### 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 ...
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### 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 $I$,$J$ ...
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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### $V$ 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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### 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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### 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 $v \in V$ ...