# Questions tagged [vector-space-isomorphism]

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

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### Computational complexity of colored vector space isomorphism over the binary field

Suppose there are two vector spaces $V$ and $W$ of the same dimension $n$ over the binary field $\mathbb{Z}_2$ together with subsets $S_V$ and $S_W$ of $V \setminus \{0\}$ and $W \setminus \{0\}$, ...
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### Is this sufficient to show the two spaces are not isomorphic?

Yesterday I was studying about isomorphisms between vecto spaces, and basically the fundamental note I highlighted stated that "two vector spaces whose dimensions are different cannot be ...
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### Does $\mathbb{R}^n\times\mathbb{R}^m$ imply $\mathbb{R}^{n+m}$ mapping?

This might be a very basic question, but does the function $$f:\mathbb{R}^n\times\mathbb{R}^m\to \mathbb{R}$$ necessarily imply the existence of a function $$g:\mathbb{R}^{n+m}\to\mathbb{R}?$$ If ...
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### Are irreducible elements of the tensor product of a vector space equivalent to irreducible polynomials?

I'm looking for some feedback on a construction I came across. Loosely, it entails sending a vector space isomorphically to a vector space of polynomials, 'restoring' the ring structure, and asking ...
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### Algebraic meaning of the roots of a polynomial in the context of linear algebra, specifically linear transformation

A polynomial of order n $p(x) = \Sigma_{i=0}^n c_i x^i$ is isomorphic to $R^n$. So is there any algebraic meaning of the roots of a polynomial in the context of linear algebra, specifically linear ...
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### Relation between $\mathbb{Q}$ and $\mathbb{Z}^2$

Is there a way of writing $\mathbb{Q}$ in terms of $\mathbb{Z}^2$ in a simple expression? My idea is to construct $\mathbb{Q}$ a quotient set of $\mathbb{Z}^2$, where $\mathbb{Z}$ is the set of ...
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### How to show that a Vector space is isomorphic with the range of a vector space

I have the following question. Let $V$ and $W$ be vector spaces, let T: $V \to W$ be a linear transformation, and let $B = \{v_1, v_2, ..., v_n\}$ be a basis of $V$. Prove that if $T$ is injective, ...
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### Proving that range(T) $\cap$ ker(T)={0} for a linear transformation T: V $\rightarrow$ V

My question is related to this other question:V is a n-dimensional vector space and $T : V \rightarrow V$ is a LT s.t $rank(T) = rank(T^2)$. Prove that $range(T)\cap ker(T) = \{\mathbb{0}\}$. I solved ...
I'd be thankful for a reference to the most general form of this theorem. If I'm not mistaken, it is true that $$\mathcal{L}(U_n, V_m) \cong M_{m\times n}(\mathbb{F}),$$ where $U_n$ is a $n$-...