# Questions tagged [vector-spaces]

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• 942
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### Which lattices have non-trivial linear representations?

Suppose we have a a bounded lattice $L$. We might ask: does there exist a non-trivial linear representation of $L$, i.e. a lattice homomorphism $\rho: L \to \text{Sp}(V)$, where $V$ is a non-zero ...
• 101
416 views

### Reducing $9\times9$ determinant to $3\times3$ determinant

Consider the $9\times 9$ matrix $$M = \begin{pmatrix} i e_3 \times{} & i & 0 \\ -i & 0 & -a \times{} \\ 0 & a \times{} & 0 \end{pmatrix}$$ for some vector $a \in \mathbb R^3$, ...
• 53
48 views

### A query regarding complex vector decomposition

Given a complex vector $V$ of length $n^2$. Each complex entry in the vector is of size (number of digits or bits required to express the complex number) $c$ for some constant $c$. Is it always ...
• 13
291 views

### A minimal semigroup generating subset of the additive reals

I asked this on MSE, but I was told to ask it here because it is a difficult question. Consider the additive magma of the real numbers, $(\mathbb{R};+)$. Does there exist a subset $S$ of the reals ...
• 2,043
83 views

### Name for closure property: set of maps closed under taking $(f,g)\mapsto (f-g)/2$

Suppose that $F$ is a collection of functions mapping some set $\Omega$ to $\mathbb{R}$, with the following closure property: whenever $f,g\in F$, we also have $(f-g)/2\in F$. Is there a name for this ...
• 5,949
106 views

### Right unitor in star-autonomous categories

1.Context Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Here $r$ denotes the right unitor. Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume ...
1 vote
98 views

• 2,073
65 views

### Field automorphisms of projective spaces without the axiom of choice

Suppose P is a projective space over the field $k$. If P has finite dimension $n$, we can fix a base. Relative to this base, the full automorphism group of P can be described by the action on the ...
• 4,015
1 vote
525 views

### Notation for the space of eventually-zero sequences

An eventually-zero sequence is a real-valued sequence $(x_n)_{n=1}^\infty$ for which there exists an $N\in\mathbb{N}$ such that $x_n=0$ for each $n\geq N$. The space of eventually-zero sequences ...
• 153
96 views

### Proving the non-existence of canonical isomorphisms

From time to time, during my undergraduate lectures on linear algebra appears the following question from the most smart students in the class. I asked to my algebra colleagues but I have not received ...
• 1,483
233 views

### Injection vs. bijection between $V^\star \otimes V^\star$ and $(V \otimes V)^\star$ [closed]

It is a well-known fact that, if $V$ is a vector space over a field $k$, then $V^\star \otimes V^\star$ embeds into $(V \otimes V)^\star$. It turns out to be an isomorphism when $V$ is a finite-...
242 views

### Application of the Frechet derivative [closed]

$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differentiable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that: \...
533 views

### Sum of $q$-binomial coefficients

Denote by $\binom{n}{k}_q = \prod_{i=0}^{k-1} \frac{ q^{n-i} - 1 }{ q^{k-i} - 1 }$, $k = 0, 1, \ldots, n$, the $q$-binomial (Gaussian) coefficients. These numbers are symmetric, in the sense ...
• 479
1 vote
113 views

### Nilpotent matrices with (Motzkin-Taussky) property L

One of the consequences of the well-known Motzkin-Taussky theorem (https://www.jstor.org/stable/1990825) is the following : if two complex matrices $A, B$ generate a vector space of diagonalisable ...
214 views

### Vector convolution?

I am working on a research problem which leads to the following optimization problem: \hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
• 263
169 views

### Looking for a paper on axiomatic orthogonality in a vector space

I am looking for a paper "Linear spaces with disjoint elements and their conversion into vector lattices" by A. I. Veksler. It was published in 1967 in Research Notes of Leningrad State ...
• 5,273
2k views

### Under which conditions: dim(W1 + W2 + W3) = dim(W1) + dim(W2) + dim(W3) − dim(W1 ∩ W2) − dim(W2 ∩ W3) − dim(W3 ∩ W1) + dim(W1 ∩ W2 ∩ W3) [closed]

Let $V$ be a finite dimensional vector space over a field $K$, and let $W_1$, $W_2$ and $W_3$ be subspaces of $V$. By analogy with the inclusion-exclusion principle for sets, and taking into account ...
1 vote
61 views

• 3,987
### Decomposition an $A$-module to irreducible ones
Let $V$ be a complex vector space (i.e, over the filed of complex numbers) and $A$ be a complex algebra. Suppose that $V$ is an $A$-module. Under what proper condition(s) there are irreducible ...