Tagged Questions

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Some wedge product computation

I want to calculate $w\wedge w$ for $w=\sum a_{ij} e_i\wedge e_j$ where the sum is over all $1\leqslant i<j\leqslant 4$. Is there a neat way to do it without writing all the terms out? What about ...
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Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
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Exterior product in 3 dimension

I'm curious about the exterior product in $3$ dimensions. Is it the same as the cross product? If it is, does anybody here have an idea about its calculation? $(a \wedge b) \wedge c = ?$
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Idempotent operators over the exterior algebra

I am wondering if there exists a (reasonably) well-known set of operators $A_i$ over the exterior algebra such that $\{A_i,A_j\} = \frac{1}{2}(A_i +A_j)$, where $\{X,Y\}=(XY+YX)/2$.
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Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$\det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ... 0answers 39 views The canonical perspective on the Hodge star operator I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ... 2answers 46 views wedge product of m vectors in \mathbb{R}^n I came across the symbol |v_1 \wedge \dots \wedge v_m|^{-1} in a paper - this is the norm of the wedge product of vectors v_k \in \mathbb{R}^n . I thought it's meaning was self-evident until I ... 0answers 80 views Norm inequality with wedge product Anyone could help me to prove this following inequality? \displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||}  where u\wedge v is the wedge ... 0answers 56 views Exterior power of a space of maps (\mathbb{K}^T) We are given a set T \neq \emptyset, \ \ p \ge 1, \ \ p_i : T \rightarrow \mathbb{K} Could you help me prove that if  \phi: (\mathbb{K}^T)^p \ni (f_1, ..., f_p) \rightarrow \rho \in ... 1answer 56 views Exterior power and alternating forms: explicit computations I would like to get a more concrete understanding of a general isomorphism I have read about. I apologize if this is too basic, but I was not satisfied with the references at my disposal. Let K be ... 1answer 307 views The Hodge *-operator and the wedge product On every Riemannian manifold M, we can consider the Hodge *-operator, which is characterised by the following formula:$$a \wedge *b = (a,b)\nu.$$Here a and b are smooth forms on M, (\ ,\ ... 0answers 26 views Exterior product of vectors in \mathbb{R}^4 with integer coefficients. Let a, b, c, d be vectors with integer coordinates in \mathbb{R}^4 such that k a \wedge b = c \wedge d for some integer k and a \wedge b \neq l v for any v \in \bigwedge^2 (\mathbb{R}^4) ... 0answers 39 views Vanishing criterion of pure wedges Let R be a commutative ring, M some R-module, and m,n \in M. Is there some criterion when m \wedge n = 0 in \Lambda^2(M)? There are some sufficient criterions, for example that m \in ... 1answer 46 views An example of the “natural” paring V^* \times V \rightarrow \mathbb{R} This so called natural paring is not natural to me at all. I am wonder if someone could give me an explicit example? I understand that V^* is the dual space of V, and to my understanding, its ... 2answers 58 views If \phi_is are linearly dependent, \det [\phi_i(v_j)] = 0 - is the proof legit? Given v_1, \ldots, v_k \in V and \phi_1, \ldots, \phi_k \in V^*. If \phi_1, \ldots, \phi_k \in V^* are linearly dependent, proof \det[\phi_i(v_j)] = 0. Here k is the dimension of V, but ... 0answers 25 views Did I give enough justification when I extend to p-dimensional? I have proved these two exercises: (1) Suppose that T \in \Lambda^p(V^*) and v_1, \ldots, v_p \in V are linearly dependent. Prove that T(v_1, \ldots, v_p) = 0 for all T \in \Lambda^p(V^*). ... 0answers 81 views Prove that \phi_1 \wedge \cdots \wedge \phi_k (v_1, \cdots, v_k) = \frac{1}{k!}\det[\phi_i(v_j)]. I have proved these two exercises: (1) Suppose that T \in \Lambda^p(V^*) and v_1, \ldots, v_p \in V are linearly dependent. Prove that T(v_1, \ldots, v_p) = 0 for all T \in \Lambda^p(V^*). ... 2answers 68 views Equality involving exterior product.. suppose you have a differential form \omega writting in local coordinates as$$\omega=\sum_{i=1}^ndx_i\wedge dy_i.$$Can anyone help me showing the following equality:$$\omega^n=n!(dx_1\wedge ...
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I recently realised that asking a question and answering our own question is allowed here, so here is a question I've seen commonly on many sites: "How does one multiply two vectors?" This is very ...
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Elements of $\wedge^2V$ expressible in the form $v_1\wedge v_2$

If $V$ is a complex vector space, then an element $w\in \wedge^2V$ is of the form $v_1\wedge v_2$ for some $v_1,v_2\in V$ iff $w\wedge w=0$ in $\wedge^4V$. Could anybody give some intuition/show why ...
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decomposable elements of $\Lambda^k(V)$

I have conjecture I have problem to prove or disprove. Let's $w \in \Lambda^k(V)$ is k-vector. $W_w=\{v\in V: v\wedge w = 0 \}$ is k-dimensional vector space if and only if $w$ is decomposable. ...
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Without choosing bases, how to show that the determinant is multiplicative in this sense?

I was recently considering this statement: Let $V$ be a finite-dimensional $k$-vector space, and let $\phi:V\to V$ be an endomorphism. Suppose that $W\subseteq V$ is a subspace that is stable ...
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2-form associated with a skew map

Given a two-form $\omega\in \Lambda^2V$ for some (say finite dimensional) vector space $V$ we may associate with $\omega$ a skew map $f_{\omega}:V\rightarrow V^*$ given by $X\mapsto \iota_X\omega$, ...
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Tensor and Wedge Product of Vectors

I have a little doubt about tensor product acting on vectors. I was reading Spivak's Calculus on Manifolds, and Spivak introduces the tensor product of multilinear functionals. Latter he introduces ...
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Determinant of the transpose via exterior products

Let $V$ be a finite-dimensional vector space over $F$ and let $\tau:V \to V$ be a linear operator. Here's my definition of the determinant: If $t:U \to U$ is a linear operator and $\dim(U)=n$ then ...
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$\wedge^k(V)^* \cong \mathrm{Alt}^k(V)$

Let $V$ be a finite dimensional real vector space, let $\mathrm{Alt}^k(V)$ denote the space of alternating $k$-linear forms on $V$ and let $\wedge^k(V)$ denote the $k^{th}$ exterior power of $V$. I ...