# Tagged Questions

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### What is the wedge product of multilinear forms?

The construction of $V^* \otimes V^*$ involves creating formal symbols and then adding in relations such as bilinearity by quotienting out. A bilinear form $V\times V\to F$ can be thought of as a ...
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### Exterior algebra of a direct sum [duplicate]

Why the exterior algebra of a direct sum of subspaces is isomorphic to a Tensorial product of it's exterior algebras, I mean, Why ⋀(V⊕W)≃⋀V⊗⋀W ?
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### Let $K$ a field, $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor. Prove that $f \wedge f =0$

Let $K$ a field, with $\operatorname{char}(K)=0$. Let $V$ a vector space over $K$, $\dim(V) \geq 1$, and be $f$ a $n$-tensor ($f \in {\mathcal T}_n(V):=\Lambda^{n}(V)$), i.e., $f$ is an multilinear ...
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Is there some precise sense in which the "alternation" functor $A$ that maps a multilinear function $f\colon M^d\to N$ to the alternating multilinear function $A(f)\colon M^d\to N$ defined by $$... 2answers 57 views ### Tensor products, existence of a unique linear map Question: Given a bilinear map B: V\times W\to X , show there exists a unique linear map T:V \otimes W\to X  s.t. B= T \circ \phi Background: We define V \otimes W  by F[ ... 3answers 93 views ### 0-th exterior power, empty product of modules and their tensor product \def\finiteprod#1#2{#1_{1}\times#1_{2}\times\dots\times#1_{#2}}In Lang's algebra he defines the tensor product as a universal object in the category of multilinear maps from \finiteprod En where ... 1answer 146 views ### Lie Derivative for Wedge Product of Vector Fields I am having trouble here. The context is: Let X, Y and S be vector fields ina a manifold (we can assume it's \mathbb{C}^2 though I'm pretty sure this should work in any manifold), and we can ... 1answer 151 views ### Alternating tensors vs p-vectors Is there a reason to differentiate between alternating tensors and p-vectors? More precisely, is the exterior algebra always isomorphic to the subalgebra of alternating tensors? Thanks. 1answer 168 views ### Multiplying vectors (answered own question) I recently realised that asking a question and answering our own question is allowed here, so here is a question I've seen commonly on many sites: "How does one multiply two vectors?" This is very ... 1answer 225 views ### Associativity property of wedge products Regarding notation, I have a bit that says: The linear function e_i^* \in V^* determined by e_i^*(e_j) = \delta_{ij} form the basis of V^*, which is called the dual basis for \epsilon = (e_1, ... 1answer 283 views ### Tensor and Wedge Product of Vectors I have a little doubt about tensor product acting on vectors. I was reading Spivak's Calculus on Manifolds, and Spivak introduces the tensor product of multilinear functionals. Latter he introduces ... 0answers 116 views ### Explicit computations of tensor and wedge product Let f\colon K^3\to K^3 be a map in Jordan canonical form having matrix$$ f=\begin{bmatrix} -1 & 1 & 0 \\ 0 & -1 & 0 \\ 0 & 0& -1 \end{bmatrix}  What is $f\otimes f$? What ...
Suppose we are given a $k$-dimensional complex vector space. Consider the second exterior power of $V$, that is $\Lambda^2V$. Denote $X=\{v_1\wedge v_2\colon v_1,v_2\in V\}$. Now I have two ...