2
votes
1answer
33 views

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 ...
3
votes
2answers
69 views

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 ...
0
votes
2answers
59 views

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? $(a \wedge b) \wedge c = ?$
1
vote
2answers
95 views

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 $\{X,Y\}=(XY+YX)/2$.
0
votes
2answers
54 views

How to prove this Gram determinant

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 \\ ...
1
vote
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. ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
4
votes
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 $M$, $(\ ,\ ...
1
vote
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 $v \in \bigwedge^2 (\mathbb{R}^4)$ ...
1
vote
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 ...
0
votes
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 ...
1
vote
2answers
58 views

If $\phi_i$s 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 ...
0
votes
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 $T \in \Lambda^p(V^*)$. ...
3
votes
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 $T \in \Lambda^p(V^*)$. ...
1
vote
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 ...
2
votes
1answer
168 views

Multiplying vectors (answered own question)

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 ...
4
votes
1answer
105 views

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 ...
2
votes
1answer
130 views

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. ...
8
votes
3answers
317 views

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 ...
1
vote
1answer
30 views

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 $X\mapsto \iota_X\omega$, ...
1
vote
1answer
283 views

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 ...
7
votes
1answer
164 views

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 ...
6
votes
1answer
162 views

$\wedge^k(V)^* \cong \mathrm{Alt}^k(V)$

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 ...
5
votes
0answers
58 views

What about other symmetric functions of the eigenvalues? [duplicate]

Possible Duplicate: Identities for other coefficients of the characteristic polynomial Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
2
votes
1answer
375 views

Interpreting the determinant of matrices of dot products

In the Euclidean space $\mathbb{R}^n$ consider two (ordered) sets of vectors $a_1 \ldots a_k$ and $b_1 \ldots b_k$ with $k \le n$. Question What is the geometrical interpretation of ...
7
votes
1answer
166 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
3
votes
1answer
134 views

What is the difference between $\ker( L \bigwedge L \overset{[-,-]}{\rightarrow} L )$ and $\langle a \wedge b \big| [a,b]=0\rangle$?

Let $L$ be a finite dimensional Lie algebra. We view the Lie bracket as a linear map on the exterior square: $$\pi:L \bigwedge L \rightarrow L$$ Define $$\bigwedge L := \langle a \wedge b \big| ...
2
votes
0answers
60 views

Determining explicitly the action on the exterior products of a vector space

Let $V$ be a 2-dimensional complex vector space with basis $e_1,e_2$. Consider the endomorphism $f:V\to V$ given by $f(e_1) = e_2$ and $f(e_2) = -e_1$ with matrix $$ \left( \begin{matrix} 0 & -1 ...
1
vote
0answers
51 views

Alternating forms tangential to a subspace.

Let $V$ be a finite-dimensional vector space with euclidean product, and let $U$ be a subspace. Now let $P$ be the projection of $V$ onto $U$, and let $\omega$ be any alternating multilinear $k$-form. ...
1
vote
0answers
93 views

Given a surjective linear mapping of free modules how do you show the corresponding matrix has an invertible minor?

The following post can is related to part c) of this problem http://www.artofproblemsolving.com/Forum/viewtopic.php?f=349&t=124137 and boils down to some issues I am having with use of wedge ...
5
votes
2answers
297 views

Using the notation of wedge product to solve a linear system of equations

I am trying to solve a problem that seems like a standard idea from linear algebra but with a the notion of wedge product and exterior algebra added it gets more complicated for someone who isn't ...
4
votes
1answer
738 views

Is there a formula for the determinant of the wedge product of two matrices?

I was going over the Wikipedia page for exterior products of vector spaces and we can define the determinant as the coefficient of the exterior product of vectors with respect to the standard basis ...
3
votes
1answer
135 views

Parametrizing the total orders on a real vector space

The following question appears trivial, but it's outside my limited experience so I'd appreciate a little feedback. A total order on a real vector space $V$ is a total ordering on its vectors which ...
6
votes
2answers
281 views

Decomposition of product of exterior products

Suppose $V$ is a $n$-dimensional vector space. What is the kernel of $$\bigwedge^p V \otimes \bigwedge^q V\longrightarrow \bigwedge^{p+q} V$$ here $p+q \le n$.
10
votes
3answers
542 views

Signs in the natural map $\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$

Let $V$ be a finite-dimensional vector space over a field $\Bbbk$. Let $V^*$ denote its dual. I strongly suspect that there is a natural map $$\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$$ that ...
5
votes
1answer
136 views

Morphism of Exterior Algebras

Let $k$ be a field, let $V$ and $W$ be $k$-vector spaces of dimensions $n$ and $m$ respectively, and let $f:V\to W$ be a $k$-linear transformation. Let $\Lambda(V)$ and $\Lambda(W)$ denote the ...
11
votes
4answers
4k views

Wedge product and cross product - any difference?

I'm taking a course in differential geometry, and have here been introduced to the wedge product of to vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) ...