# Questions tagged [exterior-algebra]

It is a quotient - of the tensor algebra, obtained by taking graded sum over whole numbers $n$ of $n$-fold tensor products - by the ideal generated by elements of the form $a\otimes a$. We write the residue class of $a\otimes b$ in this algebra, as $a\wedge b$ and call it the wedge product.

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### Degree of elements in exterior algebra

How one determines the degree of an element in the exterior algebra $\bigwedge V$, for a graded vector space V. e.g.Is it true that $\wedge^n V^2=0$, as elements of $V^2$ are of degree 2? In ...
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### Calculating exterior powers

I am confused with an example I found, which says that for a given graded vector space $V$ such that $V^0=0$ and $V^1=0$, then $(\bigwedge V)^2=\bigwedge^2V =V^2$. Two questions: 1.$V^0$ is not the ...
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### Exterior derivative independence from coordinate systems

In the book Mathematical Methods of Classical Mechanics by V.I. Arnold, the author introduces (p.189) the concept of exterior derivative as "the principal linear part of the increment" of the function ...
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### Are vector bundles with isomorphic determinant bundles isomorphic?

Let $A$ and $B$ be $2n$-dimensional complex vector bundles and $\det A=\Lambda^{2n}(A)$ and $\det B=\Lambda^{2n}(B)$. Can you prove $A\cong B$ if and only if $\det A\cong \det B$? Is it a correct ...
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### What is the covariant derivative of a wedge product?

If we have a covariant derivative for vector fields given by an affine connection on a manifold, can we extend that to a covariant derivative for k-vectors by assuming that the product rule hold for ...
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### Minimal embedding of the Grassmannian into Euclidean (or projective) space

Let $Grass(r,k)$ be the set of all $r$-dimensional subspaces of $\Bbb R^k$. It is well known that $Grass(r,k)$ embeds isometrically as a projective variety into the projectivization of the r'th power ...
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### Inner product of $k$-forms

I'm working on the following problem from Lee's Introduction to Riemannian Manifolds: Let $(M,g)$ be a Riemannian $n$-manifold. show that for each $k=1,\ldots, n$, there is a unique fiber ...
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### A way to define the wedge product

There are various equivalent ways of introducing the wedge product. One possibility is to first define $\,\wedge\,$ for basis forms,  e^{i_1}\,\wedge .\,.\,. \wedge\,e^{i_r}\;\equiv\;r!\,\left[\,e^...
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### Does every eigenspace of the exterior power $\bigwedge^k A$ corresponds to an invariant subspace?

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$ be fixed. Given an automorphism $A \in \text{GL}(V)$, consider its $k$-th exterior power $\bigwedge^k A \in \text{GL}(V)$. ...
### Abtract explanation for $\det(A+B) = ∑_{k=0}^n \langle \Lambda^k A, \Lambda^{n-k} B \rangle$
Let $A$ and $B$ be $n×n$ matrices. If we expand the determinant of $A+B$ as a sum over all permutations of $[\![1,n]\!]$ and all choices of whether the coefficient comes from $A$ or from $B$, this ...
Let $V$ be a real $d$-dimensional vector space, and let $1<k<d$ be fixed. Let $v_i$ be a basis for $V$. Consider the induced basis for the $k$-th exterior power $\bigwedge^k V$, given by \$v_{i_1}...