# Questions tagged [vector-space-isomorphism]

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

356 questions
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### 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 ...
2answers
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### 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 ...
1answer
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### 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 ...
1answer
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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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### Find the matrix of the given linear transformation T with respect to the basis and determine if it is an isomorphism

$T(x+iy)=x−iy$ from $C$ to $C$, with respect to the basis = $(1 + i, 1 − i)$
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### 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 $\frac{\mathbb{C}[x]}{<x^n-1>}$