Questions tagged [vector-spaces]
For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars
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How do I find the common invariant subspaces of a span of matrices?
Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices.
Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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Minimizing the sum of cosines of non-obtuse angles formed by $n\geq4$ concurrent lines in $3$D space
Suppose I have two lines in $3$D space passing through the origin. The smallest angle formed between them would be between $0$ and $\pi/2$. Minimizing the cosine of this angle we'll get $\cos {(\pi/2)}...
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Volume of $n$-dimensional spherical orthant in upper diagonal halfspace
Consider an $n$-dimensional Euclidean Space. Consider orthants in that space. Each orthant occupies $\frac{1}{2^n}$ of the volume of an $n$-dimensional unit sphere. Let's call that a spherical ...
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Identities for subspaces and linear maps
Wikipedia has a nice list of identities for how intersections, unions, and complements interact with images and preimages of set functions.
But if $f:V \to W$ is a linear map, many of the identities ...
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The topology of $GL(V)$
Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
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Axler "Linear Algebra Done Right" Exercise 6.B.13
This exercise appears in Section 6.B "Orthonormal Bases" in Linear Algebra Done Right by Sheldon Axler. Inner product spaces, norms, orthogonality, and orthonormal bases have been ...
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New norm with strictly coarser induced topolgy
Let $(V,\|\cdot\|)$ be an infinite dimensional normed space.
Does there alway exist a norm $|||\cdot|||$ on $V$ which induces a strictly coarser topology than $\|\cdot\|$?
I know, that there is ...
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What is $\ \overline{\bigcup_{p≥ 1}\ \{A\in M_n(\mathbb C), \ A^p = I_n\}} \ $?
Let $\Gamma_p = \{A\in M_n(\mathbb C), A^p = I_n\}$ and let $\Gamma = \bigcup_{p≥ 1}\ \Gamma_p$.
What is the closure of $\Gamma$ ? (This is from an oral exam).
Let $B \in M_n(\mathbb C)$ such that ...
user371663
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How to visualize cotangent spaces.
I was wondering how to intuitively and visually understand dual vector spaces and one-forms. So my question is (1), how to visualize cotangent spaces and (2), how to intuitively understand them? My ...
user207766
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The "semi-symmetric" algebra of a vector space
If $V$ is a vector space over a field $K$ then the symmetric algebra $S(V)$ is defined as the tensor algebra $T(V)$ factorized by the two-sided ideal generated by $x\otimes y-y\otimes x$, with $x,y\in ...
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Bases in vector spaces without $AC$
It is known that without the axiom of choice, not every vector space has a basis.
But I was wondering, if I don't assume the axiom of choice, and I choose a vector space $V$ which does have a basis (...
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Geometric interpretation of duality in optimization
There are several beautifully written posts on stackexchange about duality. For example:
A technical explanation of duality that attempts to offer some intuitions including the insight that the ...
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Is a group with transitive automorphisms necessarily a vector space?
Let $G$ be a group with transitive automorphisms on $G-\{e\}$. I.e. for any $a,b\neq e$ in $G$, there exists some $f \in \operatorname{Aut}(G)$ such that $f(a) = b$. Is it then necessarily the case, ...
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Why aren't derivatives defined on metric spaces?
I'm studying the book by Ambrosio on gradient flows in metric spaces. It's stated that the usual notion of gradient flow, $$ x'(t) = -\nabla_x f(x),$$ is not defined on metric spaces because we don't ...
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Expectation as an Operator vs. as a Functional
I've been reading probability formalisms, and I see people referring to expectation often as an operator, and less often as a functional (in the context of a vector space of random variables).
I can ...
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