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4

First of all, notice that exterior powers commute with base change (Eisenbud, Commutative Algebra with a View..., Proposition A2.2, p. 576), hence $$\Lambda^n_{F[x]} (F[x] \otimes_F V)=F[x] \otimes_F \Lambda^n_{F}V$$ You can easily check, that the following diagram (of $F$-modules) commutes ($m(\lambda)$ is the map $x \mapsto \lambda$ from the other ...

4

There isn't much you need to do. The wedge product satisfies the relationship: $\alpha \wedge \beta = (-1)^{pq} \beta \wedge \alpha$ if $\alpha, \beta \in \Lambda^p, \Lambda^q$ respectively. In your case $$\omega \wedge \omega = (-1)^{(2q+1)(2q+1)} \omega \wedge \omega = -\omega \wedge \omega$$ That can only happen if $\omega \wedge \omega = 0$.

4

I don't know what kind of answer you're expecting at this level of generality. $\wedge^k V$ is irreducible as a representation of $GL(V)$, so in some sense there is no additional decomposition that can be done knowing nothing about the group $G$. Given the character of $V$, you can compute the character of $\wedge^k V$. For example, $$\chi_{\wedge^2 V}(g) ... 3 If$$\omega = \sum_{i=1}^n\frac{(-1)^{i-1}x_i}{\|x\|^n}\,dx_1 \wedge\cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n,$$then:$$d\omega = \sum_{i=1}^n\sum_{j=1}^n\frac{\partial}{\partial x_j}\left(\frac{(-1)^{i-1}x_i}{\|x\|^n}\right) dx_j \wedge dx_1 \wedge\cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n.$$Now, the only surviving term is when j = ... 3 The Lie bracket is an alternating bilinear map. This means you can describe it in three different ways: As a function \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g} (which is alternating and bilinear), As a linear map \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g} (which is alternating), or As a linear map \Lambda^2 \mathfrak{g} \to ... 3 The easiest way to answer your question is first bring it in an 'equivalent' form and then answer it. Of course you might still argue whether the the reversed form is really equivalent... The modified question is: Can we identify e_1 \wedge \cdots \wedge \hat{e}_i \wedge \cdots \wedge e_N with e_i^\perp? (So I only moved the \perp to the other side ... 3 This is not really a direct answer but is just too long for a comment. One curious fact about the wedge construction is that \bigwedge^n V can be (functorially) realized either as a subspace of \bigotimes^n V or as a quotient. (These realizations are canonically isomorphic when the characteristic of the underlying field is 0 or greater than n, but ... 3 Since the product is alternating,$$\color{red}{e_1}\wedge\color{blue}{e_1} = -\color{blue}{e_1}\wedge\color{red}{e_1}$$and therefore$$e_1\wedge e_1 = 0$$3 This follows from the fact that \bigwedge^n (T^*M) (in other words, the space of n-forms on an n-dimensional manifold) is 1-dimensional. Since the determinant is one such form, all others are scalar multiples of it. 3 This is true if and only if the normal bundle to \xi is trivial. In one direction, if N(\xi) = TM/\xi is trivial, the bundle map TM \to N(\xi) \to \Bbb R defines a 1-form \alpha as desired (recalling that T^*M = \text{Hom}(TM,\Bbb R) as vector bundles). On the other hand, given such an \alpha, put a Riemannian metric on M and consider the ... 3 They both may be right. If wedge product differs (Similar problem) and we set the definition of exterior derivative as$$d\omega=\sum_{I}d\omega_I\wedge dx^I,$$then d may differs as well (cause \wedge appears). If we take axiomatic approach to exterior derivative, then one of axioms says ... 2 The \mathbb{Z}_2 grading is easy enough to anticipate. Given an integer it is either even or odd. So, there's your grading. A one-form is odd. A two-form is even. Even elements commute with all other elements under the wedge product whereas the product of odd elements anticommute. All of this is plainly seen in:$$ \alpha \wedge \beta = (-1)^{pq} \beta ...

2

Fix a basis $e_i$ for $V$. Let $v = \Sigma c_{ij} e_i \otimes e_j$ be a tensor in your space. If $v$ is an -1 eigenvector, then $v + Tv = 0$. But $v + Tv = \Sigma (c_{ij} + c_{ji}) e_i \otimes e_j$, which implies that $c_{ij} = - c_{ji}$. Hence $v = \Sigma c_{ij} (e_i \otimes e_j - e_j \otimes e_i)$.

2

First distribute all terms in the product defining $\omega \wedge \cdots \wedge \omega$ (there are $n$ products). Notice that all of the terms with repeated $x_j$ will vanish by antisymmetricity of the wedge product. Thus you get $$\omega^{\wedge n} = \sum_{\sigma \in S^*_n} e_{\sigma(1)}\wedge e_{\sigma(1)+1} \wedge \cdots \wedge e_{\sigma(n)}\wedge ... 2 Write A as a product of the form E_1E_2\cdots E_kDF_1F_2\cdots F_\ell, where each E_i (resp. F_j) is an elementary matrix corresponding to row (resp. column) switching or row (resp. column) addition, and D is a diagonal matrix. You are done if you can explain why \det A=\det(E_1)\cdots\det(E_k)\det(D)\det(F_1)\cdots\det(F_\ell) and the ... 2 It must be something like (-1)^{\text{parity of the permutation}}, where the parity of the permutation can be defined (there are several possible definitions) as the parity of the number of transpositions into which the permutation is decomposed. One shows that the number of factors of any two such decompositions has the same parity. Another possible ... 2 Yes, this is true. First, by hypothesis I is generated by the forms \omega_1, \dots, \omega_n, which I guess are homogeneous, thus it'll always be a graded ideal. So I is a differential ideal iff d\omega \in I for all \omega \in I. If I is a differential ideal, then necessarily d\omega_i \in I for all i; but by hypothesis, I is generated ... 2 Fix a scalar \lambda\in F. There is an evaluation map m_\lambda:F[x]\rightarrow F such that x\mapsto \lambda. It induces a commutative diagram \require{AMScd} \begin{CD} \Lambda^n_{F[x]} M @>{\Lambda^n_{F[x]}(1\otimes T-x\otimes \id_V)}>> \Lambda^n_{F[x]} M\\ @V{m_\lambda}VV @V{m_\lambda}VV \\ \Lambda^n_F V @>{\Lambda^n_F (T-\lambda ... 2 Let us write Te_i = \sum_j a^j_i e_j. For 1 \leq i < j \leq n, we have$$ (\Lambda^2(T))(e_i \wedge e_j) = Te_i \wedge Te_j = \left( \sum_{k_1} a_i^{k_1} e_{k_1} \right) \wedge \left( \sum_{k_2} a_j^{k_2} e_{k_2} \right) = (a_i^i a_j^j - a_i^j a_j^i) (e_i \wedge e_j) + \cdots $$where the \cdots don't involve e_i \wedge e_j (as the coefficient ... 2 So let's assume that V has a non-degenerate bilinear form \langle\cdot,\cdot\rangle with a basis e_1,\dots,e_n such that \langle e_i,e_j\rangle = \delta_{ij}, the Kronecker delta. Let * denote the Hodge star operator. Note that we have the formula$$ \langle x,y\rangle = *((*x)\wedge y) .$$Let's identify any operator on V with its matrix ... 2 I'll propose to you another (slightly different, but isomorphic) definition of the adjugate (classical adjoint). Im borrowing from section 8 of http://people.reed.edu/~jerry/332/27exterior.pdf . Let f:V\rightarrow V (with n the dimension of V). We have a canonical isomorphism \phi:V=\wedge^1 V\rightarrow\mathrm{Hom}(\wedge^{n-1} V,\wedge^n V) ... 2 The cotangent bundle of M \times F, as a vector bundle, is the direct sum of the cotangent bundles of M and F (more precisely, of their pullbacks along the natural projections), so you're asking how to describe the exterior powers of a direct sum V \oplus W of two vector bundles. The answer is that the exterior algebra is a graded tensor product ... 2 \bigwedge^r T^*(M\times F)=\sum_{s+t=r}\bigwedge^s T^*M\otimes\bigwedge^t T^*F 2 The wedge product is defined to be the asymmetric tensors$$ v_1 \wedge v_2 = v_1 \otimes v_2 - v_2 \otimes v_1 $$Therefore e_1 \wedge e_1 = 0 vanishes. 2 We have$$e_1\wedge e_1=-e_1\wedge e_1,$$so e_1\wedge e_1=0. 2 It is the second term that is always zero (because d^2a = 0), no matter what k is. The first term is zero because da has odd degree. In general$$ x \wedge y = (-1)^{|x||y|} y \wedge x$$where |x|,|y| are the degrees of x,y. In particular, if x has odd degree then x \wedge x = 0. 2 No, no, every 2-vector can be written as the wedge product of two vectors. For example, assuming c\ne 0, you can write$$ae_1\wedge e_2+ be_1\wedge e_3+ce_2\wedge e_3 = (be_1+ce_2)\wedge (-\frac ace_1+e_3).

2

The exterior algebra (generated by a vector space $V$) is an algebra. It is the quotient of the tensor algebra $TV$ by the relation $xy+yx=0$ for all $x,y\in V$. I think you are confusing what is the algebra and what is the vector space. $V$, in this context, does not form an algebra under the wedge product. However, the vector space $\Lambda(V)$ does. We ...

2

If $A$ is diagonalizable, so that there exists an invertible matrix $C$ such that $D=CAC^{-1}$ is diagonal, then $D=D^t=(CAC^{-1})^t=C^{-t}A^tC^t$. Since $D$ and $D^t$ have the same determinant, simply because the two matrices are in fact equal, it follows at once from this that $A$ and $A$ and $A^t$ have the same determinant. As diagonalizable matrices are ...

2

We do, but for historical reasons they are called (linear) isometries.

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